MEASUREMENTS In class VI, we have learnt about the concepts and formulae for nding the perimeter and area of of simpl simplee closed closed gures like rectangle , square square and right triangle. In this chapter, we will learn about the area of some more closed gures such as triangle, quadrilateral, parallel parallelogram, ogram, rhombus, trapezium and circle circle..
4.1 Revision Let us recall what we have learnt about the area and perimeter of rectangle, square and right triangle tr iangle..
Perimeter When we go around the boundary of the clo closed sed gure, the distance covered by us is called the perimeter per imeter..
Fig. 4.1
Perimeter of of the recta rectangle ngle = 2 × (length) + 2 × (brea (breadth) dth) = 2 [length + breadth] Perimeter of the rectangle = 2 (l + b) units where l = length, b = breadth Perimeter of the square = 4 × length of its side = 4 × side Perimeter of the square = 4 a units where a = side Perimeterr of the tr Perimete triangle iangle = Sum of the sides of the tr triangle iangle Perimeter of the triangle = (a + b + c) units where a, b, c are the sides of the triangle tr iangle
115
Chapter 4 Area The surface sur face enclosed enclosed by a closed gure is called its area.
Fig. 4.2
Area of of the the rectangle rectangle = length × breadth breadth Area of of the rectangle = l × b sq. units Area of the square = side × side Area of of the square = a × a sq. units Area of the right right triangle =
1 2
Area of the right right triangle =
1 # ^b # hh 2
× product of the sides containing 90° sq. units
where b and h are adjacent sides of the right angle. )
Find the area and perimeter of your class room blackboard, black board, table and and windows. windows.
)
Take a sheet of paper, cut the sheet into different measures of rectangles, squares and right triangles. Place them on a table and nd the perimeter and area of each gure.
Ex ample 4.1 4 .1
Find the area and the perimeter of a rectangular eld of of length 15 15 m and breadth 10 10 m. Solution
Given: length = 15 m and breadth = 10 m Area of the rectangle = length × breadth = 15 m × 10 m = 150 m2
116
Fig.4.3
Measurements Perimeter of the rectangle = 2 [length + breadth breadth]] = 2 [15 [15 +1 +10] 0] = 50 m \
Area of the recta rectangle ngle = 15 150 0 m2
Perimeter of the rectangle = 50 m Ex ample 4.2 4 .2
The area of a rectangular garden 80m long is 3200sq.m. Find the width of the garden. Solution
Given: length = 80 m, Area = 3200 sq.m leng ngth th # br brea eadt dth h Area of the rectangle = le
breadth = = `
area length 3200 80
= 40 m
Width of the garden = 40 m
Ex ample 4.3 4 .3
Find the area and perimeter of a square plot of length 40 m. Solution
Given the side of the square plot = 40 m Area of the square = side × side = 40 m × 40 m = 1600 sq.m Perimeter of the square = 4 × side
Fig. 4.4
= 4 × 40 = 160 m \
Area of the square = 1600 sq.m
Perimeter of the square = 160 m Ex ample 4.4 4 .4
Find the cost of fencing a square ower garden of side 50 m at the rate of ` 10 per metre. Solution
Given the side of the ower garden = 50 m For nding the cost of fencing, we need to nd the total length of the boundary (perimeter) and then multiply it by the rate of fencing.
117
Chapter 4 Perimeter of the square ower garden
4 × side
=
4 × 50
=
200 m
= ` 10 10
cost of fencing 1m `
=
(given)
= ` 10 10 × 200
cost of fencing 200m
= ` 2000 2000 Ex ample 4.5 4 .5
Find the cost of levelling a square park of side 60 m at ` 2 per sq.m. Solution
Given the side of the square park = 60 m For nding the cost of levelling, we need to nd the area and then multiply it by the rate for levelling. Area of of the square park = side × side = 60 × 60 = 3600 sq.m cost of levelling 1 sq.m = ` 2 `
cost of levelling 3600 sq.m = ` 2 × 3600 = ` 7200 7200
Ex ample 4.6 4 .6
In a right triangular ground, the sides adjacent to the right angle are 50 m and 80 m. Find the cost of cementing the ground at ` 5 per sq.m Solution
For nding the cost of cementing, we need to nd the area and then multiply it by the rate for cementing. Area of right triangular ground =
1 2
×b×h
where b and h are adjacent sides of the right anlges.
= 1
2
#
Fig. 4.5
(50 m # 80 m)
= 2000 m2 cost of cementing one sq.m = ` 5 `
cost of of cementing 2000 sq.m = ` 5 × 2000 = ` 10000 10000
118
1 are = 100 m2 1 hecta hectare re = 100 are (or) = 10000 m2
Measurements 4.2 Ar Area ea of Combined Plane F igur es In this section we will learn about the area of combined combined plane plane gures such as rectangle, square and right triangle taken two at a time. t ime. A villager owns two pieces of land adjacent to each other as shown in the Fig.4.6. Fig.4.6. He did not know the area a rea of land he owns. One land is in the form of rectangle of dimension 50 m × 20 m and the other land is in the form of a
Fig. 4.6
square of side 30m. Can you guide the villager to nd the total area he owns? Now,, Valarmathi and Malarko Now Malarkodi di are the leaders of Mathematics club in the school. They decorated the walls with pictures. First, Valarmathi Valarmathi made a rectangular picture of length 2m and a nd width 1.5m. 1.5m. While Malarko Malarkodi di made a picture in the shape of a right triangle as in Fig. 4.7 4. 7. The adjacent sides that tha t make the right angle a ngle are 1.5m
Fig. 4.7
and 2m. Can we nd the total decorated area? Now,, let us see some Now some examples examples for combined combined gures gures Ex ample 4.7 4 .7
Find the area of the adjacent gure:
Fig. 4.8
Solution
Area of square (1 (1) =
3 cm # 3 cm
= 9 cm2
Area of recta rectangle ngle (2) (2) = 10 cm × 4 cm = 40 cm2 `
Total area of the gure (Fig. 4.9) = ( 9 + 40 ) cm 2 = 49 cm2
Fig. 4.9
Aliter:
Area of recta rectangle ngle (1 (1) = 7 cm × 3 cm = 21 cm2 Area of of recta rectangle ngle (2 (2)) = 7 cm × 4 cm = 28 cm2 `
Total area of the gure (Fig. 4.10) = ( 21 + 28 ) cm 2 = 49 cm2
119
Fig. 4.1 4 .10 0
Chapter 4 Ex ample 4.8 4 .8
Find the area of the following gure:
Fig. 4.1 4 .11 1
Solution
The gure contains a rectangle and a right triangle
Fig. 4.1 4 .12 2
Area of the rectangle (1 (1) = 5 cm × 10 cm = 50 cm2 Area of of the right tr triangle iangle (2 (2)) = 1
2
= ` Total
#
35 2
(7 cm # 5 cm)
cm2 = 17.5 cm2
area of the gure = ( 50 + 17.5 ) cm2 = 67 67.5 .5 cm 2 Total are areaa = 67 67.5 .5 cm 2
Ex ample 4.9 4 .9
Arivu bought a square plot of side 60 m. Adjacent to this Anbu bought a rectangular plot of dimension 70 m
#
50 m. Both paid the same amount. Who is
beneted ? Solution
Fi.g 4.13
120
Measurements Area of the square plot of Ar Arivu ivu (1) (1) =
60 m # 60 m = 3600 m
2
Area of the rectangular rectangu lar plot of Anbu (2 (2)) =
70 m # 50 m = 3500 m
2
The area of the square plot is more than the rectangular rect angular plot. So, Arivu is beneted. Take two square sheets of same area. Cut one square sheet along the diagonal. How many right triangles do you have? What can you say about their area? Place them on the other square sheet. Observe and discuss. Now, take two rectangular sheets of same dimensio Now, dimensions. ns. Cut one rectangular sheet along the diagonal. How many right triangles do you have? What can you say about their area? Place them on the other sheet. What is the relatio relationship nship between the right triangle tr iangle and the rectangle? Exercise 4.1
1. Find the area of the following gures:
2. Sibi wants to cover cover the oor of of a room 5 m long and width 4 m by square tiles. If area of each square tiles is 1 m2 , then nd the number of tiles 2 required to cover the oor of a room. 3. The cost of a right triangular land and the cost of a rectangular land are equal. Both the lands are adjacent to each other. other. In a right triangular land the t he adjacent sides of the right angles are 30 m and 40 m. The dimensions of the rectangular land are 20 m and 15 m. Which is best to purchase? 4. Mani bought a square plot of side 50 50 m. Adjacent to this Ravi bought bought a rectangular plot of length 60 m and breadth 40 m for the same price. Find out who is beneted and how many sq. m. are more for him? 5. Which has larger larger area? A right triangle triangle with the length of of the sides sides containing containing the right angle being 80 cm and 60 cm or a square of length 50 cm.
121
Chapter 4 4.3 Ar Area ea of Tr Triangle iangle The area of a right triangle is half the area of the rectangle that contains it. The area of the right r ight triangle
(or)
=
1 (Product of the sides containing 90 0) 2
=
1 2
Fig. 4.14
b h sq.units
where b and h are adjacent sides of right angle
In this section we will learn to nd the area of tr iangles iangles..
To nd the area of a triangle Take a rectangular rectang ular piece of paper. paper. Name the vertices as A, B, C and D. Mark any point E on DC. Join AE and BE. We get a triangle ABE inscribed in the rectangle ABCD as shown in the Fig. 4.15 (i)
Fig. 4.1 4.15 5
Now mark a point F on AB such that that DE = AF. AF. join EF EF. We We observe that EF = BC. We call EF as h and AB as b. Now cut cut along the lines AE and BE BE and superpose two triangles (2) (2) and (3) (3) on ABE as shown in the Fig. 4.15 (iii). `
Area of D ABE = Area of D ADE + Area of D BCE
..... (1 (1))
Area of of Rectangle ABCD = Area of of D ABE + (Area of D ADE + Area of D BCE)
= Area of D ABE + Area of D ABE (By using (1)) = 2 Area of D ABE (i.e.)) 2 Area of D ABE = Area of the rectangle ABCD (i.e.
122
Measurements `
`
Area of the triangle tr iangle ABE =
1 2
(area of rectangle ABCD)
=
1 2
(length × breadth)
=
1 2
bh sq.units
Area of any triangle =
1 2
bh sq.units
Where b is the base and h is the height of the triangle.
Fig. 4.1 4 .16 6
Consider an obtuse angled triangle ABC. The perpendicular drawn from C meets the base BA produced at D. What is the area of the triangle t riangle?? Fig. 4.17
Paper folding method
Take a triangular piece of paper. Name the vertices as A, B and C. Consider the base AB as b and altitude by h. Find the midpoint of AC and BC, say D and E respectively. Join D and E and draw a perpendicular line from C to AB. It It meets at F on on DE DE and G on AB. We We observe that CF = FG.
Fig. 4.1 4 .18 8
Cut along DE and again cut it along CF to get two right triangles. tr iangles. Now, Now, place the two right triangles beside the quarilateral ABED as shown in the Fig. 4.18 (iii). Area of gure (i) = Area of gure (iii) (i.e.)) Area of (i.e. of the tr triangle iangle = Area of of the recta rectangle ngle = b =
#(
1 h) sq. units 2
1 bh 2
sq. units.
123
[CF + FG = h]
Chapter 4 Ex ample 4.1 4 .10 0
Find the area of the following gures:
Fig. 4.1 4 .19 9
Solution
(i) Given:
Base = 5 cm, Height = 4 cm
Area of the triangle PQR = =
1 bh 2 1 # 5 cm # 4 cm 2
= 10 sq.cm (or (or)) cm 2 (ii) Given:
Base = 7cm, Height = 6cm
Area of of the the triangle ABC = =
1 bh 2 1 # 7cm # 6cm 2
= 21 sq.cm (or) (or) cm2 Ex ample 4.1 4 .11 1
Area of a triangular t riangular garden is 800 sq.m. The height height of the garden is 40 m. Find the base length of the garden. Solution
Area of the triangular garden = 800 sq.m. (given (given)) 1 bh 2
1 # b # 40 2
= 800 = 800
20 b = 800 b = 40 m `
Base of the garden is 40 m.
124
(since (s ince h = 40)
Measurements Exercise 4.2
1. Find the area of the following triangles:
(i)
2. Find the area of the the (i) base = 6 cm, (ii) base = 3 m, (iii) base = 4.2 m ,
(ii)
(iii)
(iv)
triangle triang le for the following following measurements: height = 8 cm height = 2 m height = 5 m
3. Find the the base of the triangle whose whose area and height height are given below: below: 2 (i) area = 40 m , height = 8 m (ii) area = 210 cm2 , height height = 21 cm (iii) area = 82.5 m 2 , height = 10 m 4. Find the height of the triangle triangle whose area and the base are given given below: (i) area = 180 m2 , base = 20 m (ii) area = 62.5 m 2 , base = 25 m (iii) area = 20 cm2 , base = 5 cm 5. A garden is in the form of a triangle. Its Its base is 26 m and height is 28 m. Find Find the cost of levelling the garden at ` 5 per m2.
4.4 Area of the Quadrilateral A quadrilateral is a closed gure bounded by four line segments such that no two line segments cross each other.
Fig. 4.20
In the abov abovee gure g (i), (ii), (iii) are quadrilaterals. g (iv) is not a quadrilateral. 125
Chapter 4 Types of quadrilateral The gure given below below shows shows the different types ty pes of quadrilateral. quadr ilateral.
Fig. 4.21
Area of the quadrilateral In a quadrilateral ABCD, draw the diagonal AC. It divides the quadrilateral into two triangles ABC and ADC. Draw altitudes BE and DF to the common base AC. Area of the quadrilateral ABCD =
Area of 3 ABC + Area of of
=
[1
=
1 2 1 2
=
2
# AC # h1 ]
#
3 ADC
1 # AC # h2 ] 2
+[
Fig. 4.22
AC # (h1 + h2)
# d # (h1 + h2)
sq. units
where d is the length leng th of the diagonal AC and
h1 and h2
are perpendiculars drawn
to the diagon d iagonal al from f rom the opposite vertices. `
Area of the quadrilateral = 1
2
# d # (h1 + h2)
126
sq.units.
Measurements Ex ample 4.1 4 .12 2
Calculate the area of a quadrilateral PQRS shown sho wn in the gure Solution
Given: d = 20cm , h
1
= 7cm, h2 = 10cm .
Area of a quadrilateral PQRS = = =
Fig. 4.23
1 # d # ^h1 + h2h 2 1 # 20 # ^7 + 10h 2
10 # 17
= 170 cm 2 `
Area of the quadrilateral PQRS = 170 cm 2.
Ex ample 4.1 4 .13 3
A plot of land is in the form of a quadrilateral, where one of its diagonals is 200 m long. The two vertices vert ices on either side of this diagonals are 60 m and 50 m away from fr om the diagonal. What is the area of the plot of land ? Solution
Given: d = 200 m,
h1
= 50 m,
h2
= 60 m
Area of the quadrilateral ABCD ABCD = =
1 # d # ^ h1 + h2h 2 1 # 200 # ^50 + 60 h 2
= 100 \
#
Fig. 4.24
110
Area of the quadrilateral = 11000 m2
Ex ample 4.1 4 .14 4
The area of a quadrilateral is 525 sq. m. The perpendiculars from two vertices to the diagonal are 15 m and 20 m. What is the length of this diagonal ? Solution
Given: Area = 525 sq. m,
h1
= 15 m,
h2
= 20 m.
Now, we have
Area of the quad quadrilate rilateral ral 1 # d # ^ h1 + h2h 2
= 525 sq.m. = 525
127
Chapter 4 1 # d # ^15 + 20 h 2 1 # d # 35 2
= 525 = 525
d =
525 # 2 35
=
1050 35
= 30 m
The length of the diagonal = 30 m.
`
Ex ample 4.1 4 .15 5
The area of a quadrilateral PQRS is 400 cm 2. Find the length of the perpendicular drawn from S to PR, if PR = 25 cm and the length of the perpendicular from Q to PR is 15 cm. Solution
Given: d = 25 cm,
h1
= 15 cm, Area = 400 cm 2
Area of a quadrilateral quad rilateral PQRS = 400 cm2 1 2
(i.e.)
× d × (SL (SL + QM) QM) = 400 where SL = h1, QM = h2 1 # d # ^h1 + h2h 2
= 400
1 # 25 # ^15 + h2h 2
= 400
15 +
`
400 # 2 25
h2
=
= 16 16
h2
= 32 – 15 15 = 17 17
#
2 = 32
Fig. 4.25
The length of the perpendicular from S to PR is 17 cm. Excercise 4.3
1. From the gure, nd the area of the quadrilateral ABCD.
2. Find the area of of the quadrilateral quadrilateral whose diagonal and heights heights are: are: (i) d = 15 cm, h1 = 5 cm, h2 = 4 cm (ii) d = 10 cm, h1 = 8.4 cm, h2 = 6.2 cm (iii) d = 7.2 cm, h1 = 6 cm, h2 = 8 cm 3. A diagonal diagonal of of a quadr quadrilateral ilateral is is 25 cm, and perpendicular on it from the opposite opposi te vertices are 5 cm and 7 cm. Find the area of the quadr ilateral. 4. The area of a quadrilateral quadrilateral is 54 cm2. The perpendicualrs per pendicualrs from two opposite opposite vertices to the diagonal are 4 cm and 5 cm. What is the length of this diagonal? 5. A plot plot of of land is in the form of a quad quadrilate rilateral, ral, where one of its diagonals is 250 m long. The two vertices on either side of the diagonal are 70 m and 80 m away. What is the area of the plot of the land? 128
Measurements 4.5 Area of of a Par allel allelogram ogram In our daily life, we have have seen many plane gures other than t han square, rectangle and triangle tr iangle.. Do you know the other plane gures? Parallelogram is one of the other plane gures. In this section we will discuss about the parallelogram parallelogram and further f urther we are going to discuss the following: How to nd the area of a land which is a parallelogram in shape ? Can a parallelogram be converted converted into i nto rectangle of equal area ? Can a parallelogram be converted converted into i nto two triangles of equal area ?
Denition of Parallelogram Take four broom sticks. Using cycle valve tube rubber, join them and form a rectangle ( see Fig. 4.26 (i))
Fig. 4.26
Keeping the base AB xed and slightly push the corner D to its right, you will get the shape as shown in Fig. Fig. 4.26 (ii). Now answer the follo following: wing: Do the shape has parallel sides sides ? Which W hich are the sides parallel to each other? Here the sides AB and DC are parallel and AD and BC are parallel. We use the symbol ‘’ which denotes “is parallel to” i.e., i.e., AB  DC and AD  BC. ( Read it as AB is parallel to DC and AD A D is parallel to BC ). So, in a quadrilateral, if both the pair of opposite sides are parallel then it is called a parallelogram. Fig.4.27.
129
Fig. 4.27
Chapter 4 Area of the parallelogram Draw a parallelogram on a graph paper as shown in Fig. 4.28 (i)
Fig. 4.28
Draw a perpendicular line from the vertex D to meet the base AB at E. Now,, cut out the triangle AED. Place the triangle AED to the other side as Now shown in Fig. 4.28 (iii) What shape do you get? Is it a rectangle? Is the area of the parallelogram parallelogram equal to the area of the rectangle formed? Yes, Area of the parallelogram = Area A rea of the rectangle formed
Fig. 4.29
We nd that the length of rectangle formed is equal to the base of the parallelogram and breadth of rectangle is equal to the height of the parallelogram. (see Fig. 4.29) `
Area of parallelogram parallelogram = Area of of rectangle = (length × breadth) sq. Units = (base × height) sq. Units Area of of parallelo parallelogram gram = bh sq. Units
Where b is the base and h is the height of the parallelogram. `
area of the parallel par allelogram ogram
is the product of the base (b) and its
In a parallelogram
corresponding height (h). Note:: Any side of a parallelogram can Note be chosen chosen as base of of the parallel parallelogram. ogram. The perpendicular dropped d ropped on that side from the opposite vertex is known as
•
the opposite sides are paral parallel. lel.
•
the opposi opposite te angles are equal.
•
the opposite sides are equal.
•
the diagonals are not equal.
•
the diagonals bisect each other.
height (altitude).
130
Measurements Ex ample 4.1 4 .16 6
Using Usin g the data given in the gure, (i) nd the area of the parallelo parallelogram gram with base AB. (ii) nd the area of of the paral parallelogram lelogram with base AD. Solution
Fig. 4.30
The area of the parallelogram parallelogram = base × height (i) Area of of paral parallelogram lelogram with base AB = base AB × height DE = 6 cm × 4 cm = 24 cm2 (ii) Area of parallel parallelogram ogram with base AD = base AD × height height FB = 5 cm × 4.8 4.8 cm = 24 cm2 Note: Here, area of parallelogram parallelogram
with base AB is equal to the area of parallelogram parallelo gram with base AD. `
we conclude that the area of
a parallelogram can be found choosing
Find the relationship between the area of the parallelogram and
any of the side as its base with its
the triangles using Fig. 4.31.
corresponding heigh height. t. Ex ample 4.1 4 .17 7
Find the area of a parallelogram whose base is 9 cm and the altitude (height) is 5 cm. Solution Fig. 4.31
Given: b = 9 cm, h = 5 cm Area of of the parallelo parallelogram gram = b × h = 9 cm × 5 cm \
Area of of the paral parallelogram lelogram = 45 cm2
131
Chapter 4 Ex am ample ple 4.18 4.18
Find the height of a parallelogram whose area is 480 cm 2 and base is 24 cm. Solution
Given: Area = 480 cm 2, base b = 24 cm Area of the parallel parallelogram ogram = 480 b×h
= 480
24 × h = 480 h `
480 24
=
= 20 cm
height of a parallelogram = 20 cm.
Ex am ample ple 4.19 4.19
The area of the parallelogram is 56 cm 2. Find the base if its height is 7 cm. Solution
Given: Area = 56 cm 2, height h = 7 cm Area of the parallel parallelogram ogram
= 56
b×h
= 56
×7
= 56
b
b `
base of a parallelo parallelogram gram
=
56 7
= 8 cm.
= 8 cm.
Ex am ample ple 4.20
Two sides of the parallelogram PQRS are 9 cm and 5 cm. The height corresponding to the base PQ is 4 cm (see ﬁgure). Find (i) area of the parallelogram (ii) the height corresponding to the base PS Solution
Fig. 4.32
(i) Area of the parallel parallelogram ogram
=
b×h
= 9 cm × 4 cm = 36 cm 2 (ii) If the base PS ( b )
=
5 cm, then
132
Measurements Area = 36 b × h = 36
5 × h = 36 36 5
h = `
= 7.2 cm.
height corresponding to the base PS is 7.2 cm.
Think and Discuss: •
Draw different parallel parallelograms ograms with equal perimeters.
•
Can you say that they have same area? Excercise 4.4
1. Choose the correct answer answer.. i) The height height of a parallelogram parallelogram whose whose area is 300 300 cm2 and base 15 cm is (A)) 10 cm (A
(B) 15 cm
(C) 20 cm
(D) 30 cm
ii) The base of a parallelogram whose area is 800 cm 2 and the height 20 cm is (A)) 20 cm (A
(B) 30 cm
(C) 40 cm
(D) 50 cm
iii) The area of a parallelogram whose base is is 20 cm and height is 30 cm is (A) 300 cm 2
(B) 400 cm 2
(C) 500 cm 2
(D) 600 cm 2
2. Find the area of each of the following parallelograms:
3. Find the the area of the parallelogram whose whose base and height height are : (i) b = 14 cm, h = 18 cm (ii) b = 15 cm, h = 12 cm (iii) b = 23 cm, h = 10.5 cm (iv) b = 8.3 cm, h = 7 cm 4. One of the sides and the corresponding corresponding height height of a parallelogram are 14 cm and 8 cm respectively. Find the area of the parallelogram. 5. A ground is in in the form of a parallelogram. Its base is 324 324 m and its its height is 75 m. Find the area of the ground. 6. Find the height of the parallelogram parallelogram which has has an area of 324 sq. cm. and a base of 27 cm.
133
Chapter 4 4.6 Rhomb R hombus us In a parallelogram if all the sides are equal then it is called rhombus. Let the base of the rhombus be corresponding height be
h
b
units and its
units.
Since a rhombus is also a parallelogram we can use Fig. 4.33
the same formula to ﬁnd the area of the rhombus. `
The area of the rhombus = b × h sq. units.
In a rhombus, (i) all the sides sides are equal (ii) opposite sides are parallel (iii) diagonal divides the rhombus rhombus into two triangles of equal area. (iv)) the diagonal bisect each other at right angles. (iv
Area of the rhombus rhombus in terms of its diagonals In a rhombus ABCD , AB  DC and BC  AD Also, AB = BC = CD = DA Let the diagonals be
d 1
( AC ) and
d 2
( BD )
Since, the diagonals bisect each other at right angles AC
=
BD and BD
=
AC Fig. 4.34
Area of the rhombus ABCD = Area of D ABC + Area of D ADC = = = = `
8 12 # AC # OB B + 8 12 # AC # OD B 1 # AC # ^OB + ODh 2 1 # AC # BD 2 1 # d1 # d 2 sq. units 2
Area of the rhombus = =
1 6 d1 # d 2 @ sq. units 2 1 × ( product of diagonals) 2
Think and Discuss
Square is a rhombus but a rhombus is not a square. 134
sq. units
Measurements Ex ample 4.2 4 .21 1
Find the area are a of a rhombus whose side is 15 15 cm and the altitude a ltitude (height ( height)) is 10cm. Solution
Given: base = 15 15 cm, height = 10 10 cm Area of the rhombus = base
= 15 cm \
height
#
#
10 cm
Area of the rhombus = 15 150 0 cm2
Ex ample 4.22 4 .22
A ower garden is in the shape of a rhombus. The length of its diagonals are 18 m and 25 m. Find the area of the ower garden. Solution
Given:
d 1
= 18 m,
d 2
= 25 m
Area of the rhombus = = `
1 # d1 # d 2 2 1 # 18 # 25 2
Area of of the ower garden = 225 m2
Ex ample 4.23 4 .23
Area of a rhombus is 150 sq. cm. One of its diagonal is 20 cm. Find the length of the other diagonal. Solution
Given: Area = 15 150 0 sq. cm, diagonal
d 1
= 20 cm
Area of the rhombus = 15 150 0 1 # d1 # d 2 2 1 # 20 # d 2 2
= 150
10 # d 2
= 150
d 2 `
= 150
= 15 cm
The length of the other diagonal = 15 cm.
Ex ample 4.24 4 .24
A eld is in the form of a rhombus. The diagonals of the elds are 50 m and 60 m. Find the cost of levelling it at the rate of ` 2 per sq. m. Solution
Given:
d 1
= 50 m,
d 2
= 60 m 135
Chapter 4 Area
= =
1 # d1 # d 2 2 1 # 50 # 6 0 2
sq. m
= 15 1500 00 sq. m = ` 2
Cost of levelling 1 sq. m `
= ` 2 × 1500
cost of levelling 15 1500 00 sq. m
= ` 3000 3000
Take ake a rrectangular sheet. Mark the midpoints of the sides and join them as shown in i n the Fig. 4.35. 4.35.
Fig. 4.35
The shaded ﬁgure EFGH is a rhombus. Cut the light shaded triangles and join them to form a rhombus. The new rhombus is identical to the original rhombus EFGH see Fig.4.36.
Fig. 4.36 `
The area of recta rectangle ngle
=
Area of a rhomb rhombus us
= = =
Area of a rhomb rhombus us
=
136
Twice the area of rhombus 1 [area of rectangle] 2 1 6 AB # B C @ 2 1 6 HF # EG @ [ see Fig. 2 1 ^d1 # d 2h sq. units. 2
4.35 ]
Measurements Exercise 4.5
1. Choose the correct answer answer.. i) The area of a rhombus (A)) (A
(B) 3 (d1
d1 # d 2
4
# d 2)
(C)) 1 (d1 (C 2
# d 2)
(D) 1 (d1 4
# d 2)
ii) The diagonals of a rhombus rhombus bisect each other other at (A)) 30° (A
(B) 45°
(C) 60°
(D) 90°
iii) The area of a rhombus whose whose diagonals are 10 cm and 12 cm is (A) 30 cm 2
(B) 60 cm 2
(C) 120 cm 2
(D) 240 cm 2
2. Find the area of rhombus whose diagonals are i) 15 cm, 12 cm iii) 74 cm, 14.5 cm
ii) 13 cm, 18.2 cm iv) 20 cm, 12 cm
3. One side of a rhombus is 8 cm and the altitude ( height height ) is 12 cm. Find the area of the rhombus. 4. Area of a rhombus is 4000 sq. m. The length of one diagonal is100 m. Find the other diagonal. 5. A eld is in the form of a rhombus. The diagonals of the eld are 70 m and 80 m. Find the cost of levelling it at the rate of ` 3 per sq. m.
4.7 Trapezium A trapezium is a quadrilateral with one pair of opposite sides are parallel. The distance between the parallel sides is the height of the trapezium. Here the sides AD and BC
Fig. 4.37
are not parallel, but AB  DC. If the non  parallel sides of a trapezium are equal ( AD = BC ), then it is known as an isosce isosceles les trapezium t rapezium.. Here
+A
=
+B
;
+C
= +D
+B
+ + C = 180°
AC = BD +A
+ + D = 180° ;
Fig. 4.38
Area of a t r apezi apezium um ABCD is a trapezium with parallel sides AB and DC measuring ‘a’ and ‘b’. Let the distance between the two parallel sides be ‘h’. The diagonal BD divides the trapezium into two triangles ABD and BCD BCD.. 137
Chapter 4 Area of the trapezium = area of D ABD + area of D BCD
`
=
1 1 # AB # h + # DC # h 2 2
=
1 # h 6 AB + 2
=
1 # h6 a + b @ 2
DC @
Fig. 4.39
sq. units 1 # 2
Area of a trapezium =
height
#
(sum of the parallel sides) sq. units
Ex ample 4.25 4 .25
Find the area of the trapezium whose height is 10 cm and the parallel sides are 12 cm and 8 cm of length. Solution
Given: h = 10 cm, a = 12 cm, Area of a trapezium = = ∴
b
= 8 cm
1 # h^a + bh 2 1 # 10 # ^12 + 8 h 2
=
5 # ^20 h
Area of the trapezium = 100 sq. cm 2
Ex ample 4.26 4 .26
The length of the two parallel sides of a trapezium are 15 cm and 10 cm. If its area is 100 sq. cm. Find the distance between the parallel sides. Solution
Given:
a
= 15 cm, b = 10 cm, Area = 100 sq. cm.
Area of the trapezium = 100 1 h^a + bh 2
= 100
1 # h # ^15 + 10 h 2
= 100
h # 25
= 200
h `
=
200 25
=8
the distance between the parallel sides = 8 cm.
Ex ample 4.27 4 .27
The area of a trapezium is 102 sq. cm and its height is 12 cm. If one of its parallel sides is 8 cm. Find the length of the other side. Solution
Given:
Area = 102 cm 2, h = 12 cm, a = 8 cm. 138
Measurements Area of a trap trapezium ezium = 102 1 h ^a + b h 2 1 # 12 # ^8 + b h 2
= 102 = 102
6 (8 + b) = 102 8 + b = 17 `
b = 17 – 8 = 9
&
length of of the other side = 9 cm
By paper folding method: In a chart paper draw a trapezium ABCD of any measure. Cut and take the trapezium separately. Fold the trapezium in such a way that DC lies on AB and crease it on the middle to get EF.
Fig. 4.40
EF divides the trapezium in to two parts as shown in the Fig. 4.40 (ii) From D draw DG
=
EF. Cut the three parts separately.
Arrange Arr ange three par ts as shown in the Fig. Fig. 4.40 (iii) (iii) The gure obtained is a rectangle rect angle whose whose length is AB + CD = and breadth is `
1 2
( height of trapezium ) =
a+b
1 h 2
Area of trap trapezium ezium = area of recta rectangle ngle as shown shown in Fig. Fig. 4.40 (iii) = length
#
breadth
= ^a + bh` 1 hj 2
=
1 h^a + b h 2
139
sq. units
Chapter 4 Excercise 4.6
1. Choose the correct answer. i) The area of trapezium trapezium is ____________ ____________ sq. units units (A)) h^a + bh (A
(B)
1 2
(C) h^a  bh
h (a + b)
ii) In an isosceles trapezium
(D)
1 2
h (a – b)
(A)) non parallel sides are equal (A
(B) parallel sides are equal
(C) height = base
(D) parallel side = non parallel side
iii) The sum of parallel parallel sides of a trapezium trapezium is 18 cm and height height is 15 cm. Then Then its area is (A) 105 cm 2
(B) 115 cm 2
(C) 125 cm 2
(D) 135 cm 2
iv) The height height of a trapezium trapezium whose sum of parallel sides sides is 20 cm and the the area 80 cm2 is (A)) 2 cm (A
(B) 4 cm
(C) 6 cm
(D) 8 cm
2. Find the area of a trapezium whose altitudes altitudes and parallel parallel sides sides are given given below: i) altitude = 10 cm, parallel sides = 4 cm and 6 cm ii) altitude = 11 cm, parallel sides = 7.5 cm and 4.5 cm iii) altitude = 14 cm, parallel sides = 8 cm and 3.5 cm 3. The area of a trapezium is 88 cm 2 and its height is 8 cm. If one of its parallel side is 10 cm. Find the length of the other side. 4. A garden is in the form of a trapezium. The parallel sides are 40 m and 30 m. The perpendicular distance distance between the parallel side is 25 m. Find the area of the garden. 5. Area of a trapezium is 960 cm 2. The parallel sides are 40 cm and 60 cm. Find the distance between the parallel sides. 4.8 Circle
In our daily life, we come across a number of objects objects like wheel wheels, s, coins, rings, r ings, bangles,giant wheel, compact disc (C.D.) What is the shape of the above said objects? ‘round’, ‘round’, ‘round’ Yes, it is round. In Mathematics it is called a circle. Now, let us try to draw a circle. Take a thread thr ead of any length and a nd ﬁx one end tightly at a point O as shown in the ﬁgure. Tie a pencil (or a chalk) to the other end and stretch st retch the thread th read completely completely to a point A. Holding the thread stretched tightly, move the pencil. Stop it when the pencil again reaches the point A. Now see the path traced by the pencil.
Fig. 4.41
140
Measurements Is the path traced t raced by the pencil a circle or a straight line? ‘Circle’ Yes, the path traced by the point, which moves at a constant distance from a
xed point on a given plane surface is called a circle.
Par ts of of a Cir cl clee The xed point is called the centre of the circle. The constant distance between the xed point and the moving point is called the radius of the circle. i.e. The radius is a line segment seg ment with one end point at the centre and the other end on the circle. It is denoted by ‘ r ’. A line segment joining any two points on the circle is called a chord.
Fig. 4.42
Diameter is Diameter is a chord chord passing through the centre centr e of the circle. It is denoted by ‘ d ’. The diameter is the longest chord. It is twice the radius.(i.e.
d = 2r )
The diameter divides the circle circle into two two equal parts. Each equal part is a semicircle. Think it: How many diameters can
The plural of radius is “radii”.
a circle have ?
All the radii of a circle are equal.
Circumference of a circle: Can you nd the distance covered by an athlete if he takes two rounds on a circular t rack. Since it is a circular track, we cannot use the ruler to nd out the distance. So, what can we do ? Take a one rupee coin.Place it on a paper and draw its outline. Remove the coin. Mark a point A on on the outline outline as shown shown in the Fig. Fig. 4.44 4.44 Take a thread and x one end at A. Now place the thread in such a way that the thread coincides exactly exactly with the outline. Cut the other end of the th read when it reaches the point A. Length of the thread is nothing but the circumference of the coin. 141
Fig. 4.43
Fig. 4.44
Chapter 4 So, the distance around a circle is called the circumference of the circle, which is denoted by ‘C’. i.e., The perimeter of a circle is known as its circumference.
Take a bottle cap or a bangle or any other circular objects and ﬁnd the circumference. If possible ﬁnd the relation between the circumference and the t he diameter of the circular obj objects. ects.
Relation between diameter and circumferen circumference ce of the circle
Draw four circles with radii 3.5 cm, 7 cm, 5 cm, 10.5 cm in your note book. Measure their circumferences using a thread and the diameter using a ruler as shown in the Fig. 4.45 given below.
Fig. 4.45
Fill up the missing values in table 4.1 4.1 and ﬁnd the ratio of the circumference to the diameter. Circle
Radius
Diameter ( d )
Circumference (C)
1
3.5 cm
7 cm
22 cm
22 7
2
7 cm
14 cm
44 cm
44 22 = 14 7
3
5 cm
10 cm
 
  
4
10.5 cm
21 cm
 
  
Table 4.1
142
Ratio
` Cd j
= 3.14 = 3.14
Measurements What do you infer from the above table?. Is this ratio
` Cd j approximately
the same? Yes ! C d
= 3.14
C = ^3.14h d
&
So, can you say that the circumference of a circle is always more than 3 times its diameter ? Yes !
In all the cases, the radio
C d
is a constant and is denoted by the Greek letter r
(read as ‘pi’ ). Its approximate value is so,
C d
=
r
&
C=
22 7
r d
or 3.14.
units
where d is the diameter of a circle.
We know that the diameter of a circle is twice the radius r . i.e., from the above formula, C =
r d
= r ^2r h
&
d
=
2r .
C = 2r r units.
The value of r is calculated by many mathematicians.
Babylonians :
r
Archemides :
3
=3
1 7
<
Greeks r
< 3 10
:
Aryabhata :
71
Now, we use
r
=
22 7
22 or 3.14 7 62838 (or) 3.1416 r = 2000 r
=
or 3.14
Ex ample 4.28 4 .28
Find the circumference of a circle whose diameter is 21 cm. Solution
Circumference Circumfer ence of a circle = =
r d
22 # 21 7
Here
r
=
= 66 cm. Ex ample 4.29 4 .29
Find the circumference of a circle whose radius is 3.5 m. Solution
Circumference Circumfer ence of a circle = =
2r r 2#
22 # 3.5 7
= 2 × 22 22 × 0.5 0.5 = 22 m
143
22 7
Chapter 4 Ex ample 4.30 4 .30
A wire of length 88 cm is bent as a circle. What is the radius of the circle. Solution
Length of the wire = 88 cm Circumference of the circle = Length of the wire
2#
2r r
= 88
22 # r 7
= 88
r = `
88 # 7 2 # 22
= 14 cm
radius of a circle is 14 cm.
Ex ample 4.31 4 .31
The diameter of a bicycle wheel is 63 cm. How much distance will it cover in 20 revolutions revolutions?? Solution
When a wheel makes one complete revolutions, Distance covered covered in one rotation = Circumference of wheel `
circumference circu mference of of the wheel = =
r d
units
22 # 63 7
cm
= 198 cm For one revolution, the distance covered covered = 19 198 8 cm `
for 20 revolutions, revolutions, the distance covered = 20 × 19 198 8 cm = 3960 cm = 39 m 60 cm
[100 [1 00 cm = 1 m]
Ex ample 4.32 4 .32
A scooter wheel makes 50 revolutions to cover a distance of 8800 cm. Find the radius of the wheel. Solution
Distance Dista nce travelled = Number of revolutions Circumference =
i.e.,
#
Dista Dis tanc ncee tra trave vell lled ed Number of revoluti tio ons 8800 50
2r r
=
2r r
= 176
144
Circumference
Measurements 2#
`
22 # r 7
= 176 176 # 7 2 # 22
r
=
r
= 28 cm
radius of the wheel = 28 cm.
Ex ample 4.33 4 .33
The radius of a cart wheel is 70 cm. How many revolution does it make in travelling a distance of 132 m. Solution
Given: `
r
= 70 cm, Distance travelled = 132 m.
Circumference of a cart wheel = 2r r =
2#
22 # 70 7
= 440 cm Distance travelled = Number of revolutions # Circumference `
Number of revolutions =
Distance travel Distance travelled led Circumference
=
132 m 440 cm
=
13200 cm 440 cm
(1 m = 100 cm, 132 m = 13200 cm)
= 30 `
Number of revolutions = 30.
Ex ample 4.34 4 .34
The circumference of a circular eld is 44 m. A cow is tethered to a peg at the t he centre of the eld. If the cow can graze the entire eld, nd the length of the rope used to tie the cow. Solution
Length Lengt h of the rope = Radius of the circle Circumference Circumfer ence = 44 m (given) Fig. 4.46
i.e., 2r r = 44 2#
22 # r 7 `
`
= 44
r =
44 # 7 2 # 22
=7m
The length of the rope used to tie the cow is 7 m. 145
Chapter 4 Ex ample 4.35 4 .35
The radius of a circular ﬂower garden is 56 m. What is the cost of fencing it at 10 ` 10
a metre ? Solution
Length to be fenced = Circumference of the circular ﬂower garden Circumference of the ﬂower garden
= 2r r = 2 # 22 7
`
#
56 = 352 m
Length Lengt h of the fence = 35 352 2m
Cost of of fencing fencing per metre = ` 10 cost of fencing 352 m = ` 10
`
#
352
= ` 3520 3520 `
Total cost of fencing is ` 3520. 3520.
Ex ample 4.36 4 .36
1100. What is The cost of fencing a circular park at the rate of ` 5 per metre metre is ` 1100. the radius of the park. Solution
Cost of fencing = Circumference `
#
Rate
Cost of fencing Rate 1100 = 5
Circumference = i.e., 2r r
2r r = 220 `
2 # 22 7
#
r
= 220
r
= 2 20 # 7 2 # 22
= 35 m `
Radius of the park = 35 m. Activity  Circular Geoboard
Take a square Board and draw a circle. circum ference of the circle. circle. ( See ﬁg ) Fix nails on the circumference Using Usin g rubber r ubber band, form various diameters, chords, radii and compare.
146
Measurements Excercise 4.7
1. Choose the correct answer: i) The line segment that joins the centre of a circle to any point on the circle is called (A)) Diamete (A Diameterr
(B) Radiu Radiuss
(C) Chord
(D) None
ii) A line segment joining any any two points on the circle is called (A)) Diamete (A Diameterr
(B) Radiu Radiuss
(C) Chord
(D) None
iii) A chord passing passing through through the centre centre is called (A)) Diamete (A Diameterr
(B) Radius
(C) Chord
(D) None
iv) The diameter of a circle is 1 m then its radius is is (A)) 100 cm (A
(B) 50 cm
(C) 20 cm
(D) 10 cm
v) The circumference circumference of a circle whose radius radius is 14 cm is (A)) 22 cm (A
(B) 44 cm
(C) 66 cm
(D) 88 cm
2. Fill up the unknown in the following table:
radius ( r )
diameter ( d )
circumference (c)
(i)
35 cm
 
 
(ii)

56 cm
 
(iii)
 
 
30.8 cm
3. Find the circumference circumference of a circle whose diametre diametre is given below: (i) 35 cm
(ii) 84 cm
(iii) 119 cm
(iv) 147 cm
4. Find the circumference circumference of a circle whose radius radius is given below: (i) 12.6 cm
(ii) 63 cm
(iii) 1.4 m
(iv) 4.2 m
5. Find the radius radius of a circle whose circumference circumference is given below: (i) 110 cm
(ii) 132 cm
(iii) 4.4 m
(iv) 11 m
6. The diameter of a cart wheel is 2.1 m. Find the distance travelled travelled when it it complets 100 revolutions. 7. The diameter of a circular circular park is is 98 m. Find the cost of fencing it at ` 4 per metre. 8. A wheel makes 20 20 revolutions revolutions to cover cover a distance distance of 66 m. Find the diameter of the wheel. 9. The radius of a cycle cycle wheel is 35 cm. How How many revolutions revolutions does does it make to cover a distance of 81.40 m?
147
Chapter 4 Area of a circle
Consider the following A farmer lev levels els a circular ﬁeld of radius 70 m. What will w ill be the cost c ost of levelling? What will be the cost of polishing a circular tabl t abletop etop of radius 1.5 m ? How will you ﬁnd the cost ? To ﬁnd the cost what do you need to
ﬁnd
actually?
Area or perimeter ? Area, area, area
The region enclosed by the circumference of a circle is a circular region.
Yes. In such cases we need to ﬁnd the area of the circular region. So far, you have learnt to ﬁnd the area of triangles and quadrilaterals quadr ilaterals that made up of straight lines. But, a cirlce is a plane ﬁgure made up of curved line different from other plane ﬁgures. So, we have to ﬁnd a new approach which which will make the circle turn into a ﬁgure with straight lines. Take a chart paper and draw a circle. Cut the circle and take it separately. Shade one half of the circular region. Now fold the entire circle into eight parts and cut along the folds (see Fig. 4.48).
Fig. 4.48
Arrange the pieces as shown below.
Fig. 4.49
What is the ﬁgure obtained? These eight pieces roughly form a parallelo parallelogram. gram. Similarly,, if we divide the circle into 64 equal parts and arrange Similarly ar range these, it gives nearly a rectangle. (see Fig. 4.50) 148
Measurements
Fig. 4.50
What is the breadth of this rectangle? The breadth of the rectangle recta ngle is the radius of the circle circle.. i.e.,
breadth b = r
..... (1)
What is the length of this rectangle ? As the whole circle is divided into 64 equal parts and on each side we have 32 equal parts. Therefo Therefore, re, the length of the rectangle is the length of 32 equal parts, which is half of the circumference of a circle circle.. `
length
l
= =
`l
=
1 [circumference 2 1 62r r @ = r r 2
of the circle]
..... (2)
r r
Area of the circle = Area of the rectangle (from (from the Fig Fig.. 4.50) 4.50) =
l#b
= (r r) # r
` Area
(from (1) and (2))
2
sq. units.
2
sq. units.
=
r r
of the circle =
r r
Ex ample 4.37 4 .37
Find the area of a circle whose diameter is 14 cm Solution
Diameter d = 14 cm So,
radius
r
=
Area of circle = =
d 2
=
14 2
= 7 cm
2
r r
22 #7#7 7
= 154 sq. cm `
Area of circle = 154 sq. cm 149
Chapter 4 Ex am ample ple 4.38
A goat is tethered by a rope 3.5 m long. Find the maximum area that the goat can graze. Solution
Radius of the circle = Length of the rope `
radius
r
= 3.5 m =
maximum area grazed by the goat = = = `
7 2 2
r r
m sq. units.
22 7 7 # # 7 2 2 77 = 38.5 sq. 2
Fig. 4.51
m
maximum area grazed by the goat is 38.5 sq. sq. m.
Ex am ample ple 4.39
The circumference of a circular park is 176 176 m. Find the area of the park. Solution
Circumference Circum ference = 176 176 m (given (given))
2#
2r r
= 176
22 # r 7
= 176
r
` r
=
17 6 # 7 44
= 28 m
Area of the park =
2
r r
=
22 # 2 8 # 28 7
=
22 # 4 # 28
= 2464 sq. m. Ex am ample ple 4.40
A silver wire when bent in the form of a square encloses an area of 121 sq. cm. If the same wire is bent in the form of a circle. Find the area of the circle. Solution
Let a be the side of the square Area of the squar squaree = 12 121 1 sq. cm. (given) 2
a
= 121
&
a
150
= 11 cm
(11 (1 1 # 11 = 121)
Measurements Perimeter of the square = =
4a
units
4 # 11
cm
= 44 cm Length of the wire = Perimeter of the square = 44 cm The wire is bent in the form of a circle The circumference of the circle = Length of the wire `
circumference of a circle = 44 cm `2#
2r r
= 44
22 # r 7 r
= 44 =
44 # 7 44
r = 7 cm `
Area of the circle = =
2
r r
22 7
× 7 cm × 7 cm
Area of the circle = 154 cm2. Ex ample 4.41 4 .41
When a man runs around circular plot of of land 10 times, the distance covered by him is 352 m. Find the area of the plot. Solution
Distance covered in 10 10 times = 352 m Distance cov covered ered in one time =
352 10
m = 35.2 m
The circumference of of the circular plo plott = Distance covered covered in one one time `
2#
circumference circu mference = 35.2 35.2 m 2r r
= 35.2
22 # r 7 r
= 35.2 =
35.2 # 7 44
=
0.8 # 7
= 5.6 m Area of of the circula circularr plot = =
2
r r
22 # 5.6 # 5.6 7
= 22 # 0.8# 5. 5.6 6 = 98.56 m2 `
Area of circula circularr plot = 98.56 m2
151
Chapter 4 Ex ample 4.42 4 .42
A wire in the shape of a rectangle of length length 37 cm and width 29 cm is reshaped in the form of a circle. Find the radius and area of the circle. Solution
Length of the wire = perimeter of the rectangle = 2 [ length + breadth breadth ] = 2 [37 [37 cm + 29 cm] cm] = 2
#
66 cm
= 13 132 2 cm. Since wire is bent in the form of a circle, The circumference of the circle = The length length of of the the wire `
Circumference Circum ference of a circle = 13 132 2
2#
2r r
= 132
22 # r 7
= 132
r
`
=
132 # 7 44
= 21 21
radius of the circle = 21 cm Area of the circle = =
`
2
r r
22 # 2 1 # 21 7
=
2 2 # 3 # 21
Area of of the circle = 13 1386 86 sq. cm. Exercise 4.8
1. Find the area of the circles circles whose diameters are given given below: (i) 7 cm
(ii) 10.5 cm
(iii) 4.9 m
(iv) 6.3 m
(take
r
=
22 7
)
(take
r
=
22 7
)
2. Find the area of the circles circles whose radii are are given below: (i) 1.2 cm
(ii) 14 cm
(iii) 4.2 m
(iv) 5.6 m
3. The diameter diameter of a circular plot plot of ground ground is 28 m. m. Find the cost of levelling levelling the the ground at the rate of ` 3 per sq. m. 4. A goat is tied to a peg on a grass land with a rope 7 m long. Find the maximum area of the land it can graze. 5. A circle and a square each have a perimeter of 88 cm. Which has a larger area? 6. A wheel goes a distance of 2200 m in 100 revolution revolutions. s. Find the area of the wheel. 7. A wire is in the form of a circle of radius 28 cm. Find the area that will enclose, if it is bent in the form of a square having its perimeter equal to the circumference of the cirlce. 8. The area of circular plot is 3850 m 2. Find the radius of the plot. Find the cost of fencing the plot at ` 10 10 per metre. 152
Measurements 4.9 Area of the path way In our day d ay  to  day life we go for for a walk in a park, or in a play ground or even around a swimming swim ming pool. Can you represent the path way of a park diagrammatically diagra mmatically ? Have you ever wondered if it is possible to nd the area of such paths ? Can the path around the rectangular pool be related to the mount around the photo in a photo frame ? Can you think of some more examples? In this section we will learn to nd • Area of of rectangular pathw pathway ay • Area of circular pathway pathway
Area of rectangular pathway (a)) Area of uniform (a uni form pathway pathway outside the rectangle rec tangle Consider a rectangular building. A uniform ower garden is to be laid outside the building. How do we nd the area of the ower garden? The uniform ower garden including the building is also a rectangle in shape. Let us call it as outer rectangle. We call the building as inner rectangle. Let l and
b
be the length and breadth of the
Fig. 4.52
building.. building `
Area of the inner rectangle rect angle = l b sq. units.
Let
w
be the width of the ower garden. Fig. 4.53
What is the length and breadth of the outer rectangle ? The length of of the outer recta rectangle ngle (L) =
w+l+w
The breadth of of the outer recta rectangle ngle (B) =
w+b+w
`
area of the outer rectangle = L
#
= ^l + 2wh units = ^b + 2wh units
B
= ^l + 2wh^b + 2w h sq. units Now,, what is the area of the ow Now ower er garden ?
153
Chapter 4 Actually, the area of the
ﬂower
garden is the pathway bounded between two
rectangles. `
Area of the ﬂower garden =
(Area of building and ﬂower garden) – (Area of building)
Generally, Area of the pathway = (Area of outer rectangle) – (Area of inner rectangle) i.e. Area of the pathway = ( l + 2w) (b + 2w) – lb. Ex ample 4.43 4 .43
The area of outer rectangle is 360 m 2. The area of inner rectangle rect angle is 280 m 2. The two rectangles have uniform pathway between them. What is the area of the pathway? pathway? Solution
Area of of the pathway
= (Area of outer recta rectangle ngle)) – (Area of inne innerr recta rectangle ngle)) = (360 – 280) m 2 = 80 m 2
`
Area of the pathway = 80 m 2
Ex ample 4.44 4 .44
The length of a building is 20 m and its breadth is 10 m. A path of width 1 m is made all around the building outside. Find the area of the path. Solution
Inner rectangle (given) l
= 20 m
b
= 10 m
Outer rectangle width,
=1m
w
L
Area = l × b
= l + 2w = 20 + 2 = 22 m
Area = 20 m × 10 m
B
= b + 2w = 10 + 2 = 12 m
= 200 m2
Area = (l + 2w) (b + 2w) Area = 22 m
#
12 m
= 264 m2 Area of the path = (Area of outer rectangle recta ngle)) – (Area (Area of inner rectangle) rect angle) = ( 264 – 200 ) m 2 = 64 m 2 `
Area of the path = 64 m 2
154
Measurements Ex ample 4.45 4 .45
A school auditorium is 45 m long and 27 m wide. This auditorium is surrounded by a varandha varand ha of width 3 m on its outside. Find Find the area ar ea of the varandha. Also, nd the cost of laying the tiles in the varandha at the rate of ` 100 100 per sq. m. Solution Fig. 4.54
Inner (given) rectangle l
= 45 m
b
= 27 m
Area = 45m
Outer rectangle Width,
w
=3m
L = #
27 m
l + 2w
= 45 + 6 = 51 m B =
= 1215 m2
b + 2w
= 27 + 6 = 33 m Area = 51m
#
33 m
= 1683 m2 (i) Area of the verandha = (Area of outer recta rectangle ngle)) – (Area of inne innerr recta rectangle ngle)) = (1 (1683 683 – 121 1215) 5) m2 = 468 m2 `
Area of of the verandha = 468 m2 (or) 468 sq. m. Cost of laying tiles for 1 sq. m = ` 100
(ii)
Cost of of laying tiles for 468 sq. m = ` 100
#
468
= ` 46,800 ` Cost
of laying laying tiles tiles in the verandha = ` 46,800
(b) Area of uniform pathway inside a rectangle A swimming pool is built in the middle of a rectangular ground leaving an uniform width all around it to maintain the lawn. If the pathway outside the pool is to be grassed, how can you nd its cost ? If the area of the pathway and cost of grassing per sq. unit is known, then the cost Fig. 4.55
of grassing the pathway can be found. 155
Chapter 4 Here, the rectangular ground g round is the outer rectangle where
and
l
b
are length and
breadth. `
Area of the ground (outer rectangle) =
lb
sq. units
If w be the width of the pathway (lawn), what will be the length and breath of the swimming pool ? The length of the swimming pool pool =
lww
=
l  2w
The breadth of the swimming swimm ing pool = = `
bww b  2w
Area of the swimming pool (inner rectangle) rectangle) = ^l  2wh^b  2wh Sq. units Area of of the lawn = Area of of the ground – Area of of the swimming pool. pool.
Generally, Area of the pathway = (Area of outer recta rectangle ngle)) – (Area of inne innerr recta rectangle ngle)) =
lb
– (l – 2w) (b – 2w)
Ex ample 4.46 4 .46
The length and breadth of a room are 8 m and 5 m respectively. A red colour border of uniform width of 0.5 m has been painted all around on its inside. Find the area of the border. Solution Fig. 4.56
Outer (given)rectangle l
=8m
b
=5m
Area = 8m
Inner rectangle width,
w
L #
5m
= 0.5 m =
l  2w
= (8 – 1) m
= 40 m2
B
=
=7m
b  2w
= (5 – 1) m
=4m
Area = 7m × 4 m = 28 m2 Area of the path = (Area of of outer outer rectangle) – (Area of of inner rectangle) = (40 – 28) 28) m2 = 12 m 2 `
Area of the border painted with red colour = 12 m 2
156
Measurements Ex ample 4.47 4 .47
A carpet measures 3 m
#
2 m. A strip of 0.25 m wide is cut off from it on all
sides.. Find the area of the remaining carpet sides car pet and also nd the area of strip str ip cut out. Solution
Outer rectangle
Inner rectangle
carpet before cutting the strip
carpet after cutting the strip
l
=3m
b
=2m
Area = 3m
width, #
w
= 0.25 m
L
=
2m
l  2w
= (3 – 0.5) m
= 2.5 m
= 6 m2
B
=
b  2w
= (2 – 0.5) m
= 1.5 m Area
= 2.5m
#
1.5m
= 3.75 m2 The area of the car pet after cutting the t he strip = 3.7 3.75 m2 Area of the strip cut out = (Area of the carpet) carpet) – (Area of the remaining part) pa rt) = (6 – 3.7 3.75) 5) m2 = 2.25 m2 `
Area of the strip cut out = 2.25 m 2
Note: If the length and breadth of the inner rectangle Note: rect angle is given, then the length and breadth of the outer rectangle is l + 2w , b + 2w respectively where w is the width of the path way. way. Supposee the length and Suppos a nd breadth of the outer rectangle is given, then the length and breadth of the inner rectangle is l  2w , b  2w respectively.
Fig. 4.57
Exercise 4.9
1. A play ground 60 m extended area.
#
40 m is extended on all sides by 3 m. What is the
2. A school play ground is rectangular in shape with length 80 m and breadth breadth 60 m. A cemented pathway running all around it on its outside of width 2 m is built. Find the cost of cementing if the rate of of cementing 1 sq. m is ` 20. 20. 3. A garden is in the form of a rectangle of dimension 30 m # 20 m. A path of width 1.5 m is laid all around the garden on the outside at the rate of ` 10 10 per sq. m. What is the total expense.
157
Chapter 4 4. A picture is painted on a card board 50 cm long and 30 cm wide such that there is a margin of 2.5 cm along each of its sides. Find the total area of the margin. 5. A rectangular hall has 10 m long and 7 m broad. A carpet is spread in the centre leaving a margin of 1 m near the walls. Find the area of the carpet. Also ﬁnd the area of the un covered ﬂoor. 6. The outer length and breadth of a photo frame is 80 cm , 50 cm. If the width of the frame is 3 cm all around the photo. What is the area of the picture that will be visible?
Circular pathway Concentric circles Circles Circl es drawn dr awn in a plane with a common centre and different dif ferent radii are called concentric concentric circles.
Circular pathway A track of uniform width is laid around a circular park for
Fig. 4.58
walking purpose. Can you ﬁnd the area of this track ? Yes. Area of the track is the area bounded between two concentric circles. In Fig. 4.59, O is the common centre of the two circles.Let circles. Let the radius ra dius of the outer circle be R and inner circle be r . The shaded portion is known as the circular ring or the circular pathway. i.e. a circular pathway is the portion bounded between two concentric circles. width of the pathway,
w
= R–
r
units
Fig. 4.59
i.e., w = R – r & R = w + r units r = R – w units. The area of the circular path = (area of of the outer circle) circle) – (area of of the inner circle) = rR
`
2

2
r r
= r ^ R
2
 r h
sq. units
^ R
2
 r h
sq. units
The area of the circular path =
r
2
2
= r ^ R + r h^ R  r h sq. units Ex ample 4.48 4 .48
The adjoining ﬁgure shows two concentric circles. The radius of the larger circle is 14 cm and the smaller circle is 7 cm. Find (i) (ii)
The area of the larger circle. The area of the smaller circle.
(iii)) (iii
The area of of the shaded region between two circles. 158
Fig. 4.60
Measurements Solution
i) Larger circle
ii) Smaller circle
R = 14 area =
r R
= 7
r 2
area =
22 # 14 # 14 7
=
= 22
#
=
28
2
r r
22 #7#7 7
= 22
= 61 616 6 cm 2
#
7
= 154 cm 2
iii) The area of the shaded region = (Area of larger circle) – (Area (Area of smaller circle) circle) = (6 (616 16 – 154) cm 2 = 462 cm 2 Ex ample 4.49 4 .49
From a circular sheet of radius 5 cm, a concentric circle of radius 3 cm is removed. Find the area of the remaining sheet ? (Take
r
= 3.14 )
Solution
Given: R = 5 cm, r = 3 cm Area of the remaining sheet =
^ R
r
2
 r h 2
= 3.1 3.14 4 (52 – 32) = 3.1 3.14 4 (25 – 9) = 3.1 3.14 4 × 16 = 50.24 cm 2 Aliter: R
Outer circle = 5 cm
Area =
r R
2
Inner circle r = 3 cm
sq. units
Area =
2
r r
sq. units
= 3.14 # 5 # 5
=
= 3.14
= 3.14
#
25
= 78.5 cm2
3.14 # 3 # 3 #
9
= 28.26 cm2
Area of the remain remaining ing sheet = (Area of outer circle) circle) – (Area of inne innerr circle) circle) = (78.5 – 28.26) cm 2 = 50.24 cm2 `
Area of the remaining sheet = 50 50.24 .24 cm2 159
Chapter 4 Ex ample 4.50 4 .50
A circular ﬂower garden has an area 500 m 2. A sprinkler at the centre of the garden can cover an area that has a radius of 12 m. will the sprinkler sprink ler water the entire garden (Take
r
= 3.14)
Solution
area of the garden = 500 m 2
Given,
Area covered covered by a sprink sprinkler ler = =
2
r r
3 . 14 # 12 # 1 2
Fig. 4.60
= 3.14
#
144
= 452 .1 .16 6 m2 Since, the area covered covered by a sprinkler sprink ler is less than the area of the circular
ﬂower
garden, the sprinkler cannot water the entire garden. Ex ample 4.51 4 .51
A uniform circular path of width 2 m is laid out side a circular park of radius 50 m. Find the cost of levelling the path at the rate of
` 5
per m 2 (Take
r
= 3.14)
Solution
Given:
r
= 50 m,
w
= 2 m, R =
Area of the circular path =
r+w
= 50 + 2 = 52 m
^ R + r h^ R  r h
r
=
3.14 # ^52 + 50 h^52 5 2  50 h
=
3.14 # 102 # 2
=
3.14 # 2 04
Fig. 4.6 4 .61 1
= 640.56 m 2 The cost of levelling levelling the path of area 1 sq m = ` 5 The cost of levelling the path of 640.56 m 2 = ` 5
#
640.56
= ` 3202.80 3202.80 `
the cost of levelling the path = ` 3202.80 3202.80
160
Measurements Exercise 4.10
1. A circus tent has a base radius of 50 m. The ring at the centre centre for the the performance by an artists is 20 20 m in radius. Find the the area left for the audience. (Take r = 3.14) 2. A circular eld of radius 30 m has a circular path of width 3 m inside its boundary.. Find the area of the path boundary path (Take (Take r = 3.14) 3. A ring shape metal plate has an internal internal radius of 7 cm and an external radius of 10.5 cm. If the cost of material is ` 5 per sq. cm, nd the cost of 25 rings. 4. A circular well has radius 3 m. If a platform of uniform width of 1.5 1.5 m is laid around it, nd the area of the platform . (Take r = 3.14) 5. A uniform circular path of width width 2.5 m is is laid outside outside a circular park of radius 56m. Find the cost of levelling the path at the rate of ` 5 per m2 (Take r = 3.14)
F igu r e
A r ea
1 2
1 2
× base × height
For u m u la
1 2
× b × h sq. units.
1 2
× d × (h1 + h2) sq.
× diagonal × (sum
of the perpendicular distances drawn to the diagonal from the
units
opposite vertices)
base × corresponding altitude
161
bh sq. units
Chapter 4
1 2
× product of diagonals
1 2
× d 1 × d 2 sq. units
Rhombus
1 2
1 2
× height × sum of parallel sides sides
× h × (a + b) sq. units
Trapezium
Perimeter of the circle = 2×
r
2 r r units
× radius
Area of the circle = r
r 2 sq. units
r
× radius × radius
Area of the pathway
Area of outer
i) area of the rectangular
rectangle – Area of
pathway pathw ay
inner rectangle
Area of outer circle – ii) area of the circular pathway pathw ay
Area of inner circle =
r
=
(R 2 – r 2) sq. units r
(R + r ) (R – r ) sq. units
Circular Pathway
162
GEOMETRY Geometry is a branch of Mathematics Mathematics that deals with the properties properties of various
geometrical shapes and gures. In Greek the word “Geometry” means “Earth Measurement”. Geometry deals with the shape, size, position and other geometrical properties of various obj objects. ects. Geometry is useful in studying space, architecture, design and engineering.
5.1.. Revision 5.1 R evision Basic Geometrical concepts: In earlier classes you have studied about some geometrical concepts. Let us recall them.
Point A ne dot made with a sharp pencil may be taken as roughly representing a point. point. A point has a position positi on but it has no length, breadth or thickness. t hickness. It is denoted by a capital capital letters. In the t he gure A, B, C, D are points.
Fig. 5.1
Line A line is traced out by a moving point. If the point of a pencil is moved over a sheet of paper, the trace left represents a line. A line has length, but it has no breadth. A line has no
Fig. 5.2
end points. A line AB is written as AB . A line may be named with small letters l, m, n, etc. we read them as line l, line m, line n etc. A line has no end points as it goes on endlessly in both directions.
R ay A ray has a starting star ting point but has no end point. point. The starting starti ng point is called the initial initial point. Here OA is called the ray and it is written as
the ray starts from O and passes through A. 163
OA .
That is
Fig. 5.3
Chapter 5 Line Segment Let
AB be
a straight line. line.
Two points C and D are taken on it. CD is a part of AB. CD is called a line segment, and is written as
CD .
Fig. 5.4
A line segment has two end points.
Plane A plane is a at surface which extends extends indenitely in all directions. The upper surface of a table, the blackboard, the walls are some examples of planes.
5.2.. Symmetr 5.2 Symmetr y Symmetry is an important importa nt geometrical geometrical concept commonly seen in nature and is
used in every eld eld of our life. life. Art Artists, ists, manufacturers, designers, architects architects and others make use of the idea of symmetry. The beehives, owers, tree leaves, hand kerchief, utensils have symmetrical design.
Fig. 5.5
Symmetry refers to the exact match in shape and size between two halves of an object. object. If we fold fold a pictu picture re in half and a nd both the halvesleft halvesleft half and a nd right half  match
exactly then we say that the picture is sym metrical. For example, if we cut an apple into two equal halves, halv es, we observe that two parts par ts are in symmetry symmetr y.
Tajmahal in Agra is a symmetrical monumen monument. t. Fig. 5.6
164
Geometry A buttery is also an example of a symmetrical form. If a line is drawn down the centre of the buttery’s body, each half of the buttery looks the same.
Fig. 5.7
Symmetry is of different types. Here we we discuss about about 1. Line of of symmetry or axis of symmetry 2. Mirror symmetry 3. Rotatio Rotational nal symmetry
1. Line of symmetr symmetr y In the Fi Fig g 5.8 the dotted lines divide the gure into two identical parts. If gure gu re is folded along the line, one half of the gure will coincide exactly with the other half. This dotted line is known k nown as line of symmetry. When a line divides a given gure into two equal halves such that the left and right rig ht halves matches exactly then we say that the gure is symmetrical about the line. This line is called the line of symmetry or axis of symmetry symmetr y.
Fig. 5.8
Activity 1:
Take a rectangular sheet of paper.. Fo paper Fold ld it once lengthwise lengthwise,, so that one half ts exactly over the other half and crease the edges. Fig. 5.9
Now open it, and again fold fold it once along its width. 165
Chapter 5 In this t his paper folding, You observe that a rectangle has two lines of symmetry. Discuss: Does a parallelogram have a line of symmetr y? Activity 2:
One of the two set squares in your geometry box has angle of 0
0
0
measure 30 , 60 , 90 . Take two such identical identical set square squares. s. Place them
side by side to form a ‘kite’ as shown in the Fig. 5.10. How many lines of symmetry does the shape have? You observe that this kite shape gure has one line of symmetry about its vertical diagonal. Fig. 5.1 5.10 0
Activity 3:
For the given given regular regula r polygons polygons nd the lines line s of symmetry symmetr y by using paper folding method and also draw the lines of symmetry by dotted lines. li nes.
Fig. 5.11
In the above paper folding, you observe that (i) An equilateral triangle has three lines of of symmetry.
A polygon is said to be regular if all its sides are
(ii) A square square has four four lines of symmetry (iii) A regular pentagon has ve lines of
of equal length and all its angles are of equal measure.
symmetry. (iv)) A regular hexagon has six lines of symmet (iv symmetry. ry.
Each r egular pol polyg ygon on has a s many lines of of symmetr y as it ha s sides sides .
166
Geometry
Identify the regular polygon A circle has many lines of
symmetry. Some objects and gures have no line of symmetry.
Make a list of English alphabets which have no line of symmetry Fig. 5.12
The above gures have no line of symmetry; because these gures are not symmetrical. We We can say that these gures are asymmetrical. asymmetr ical. To reect an object means to produce its mirror image.
Mir ror li line ne symme symmetr tr y When we look into a mirror we see our image is behind the mirror. This image is due to reection reection in the the mirror. We know that the image is formed as far behind the mirror as the object is in front Fig. 5.13
of it. In the above gure if a mirror is placed along the line at the
middle, the half part of the gure reects through the mirror creating the remaining identical identi cal half. In other words, the line were the mir ror is placed divides the gure into two identical parts par ts in Fig. 5.1 5.13. 3. They are of the same size and a nd one side of of the line will have its reection exactly at the same distance on the other side. Thus it is also known as mirror line symmetry. While dealing with mirror reection, we notice that the leftright changes as seen in the gure. Ex ample 5.1
The gure shows the reection of the mirror lines.
167
Chapter 5 Exercise 5.1
1. Choose the correct answer. i) An isosceles triangle has
(A)) no lines of symmetr (A symmetry y
(B) one line of symmetr symmetry y
(C)) three lines of symmetr (C symmetry y
(D) many lines of symmetr symmetry y
ii) A parallelogram has
(A)) two lines of symmetr (A symmetry y
(B) four lines of symmetr symmetry y
(C)) no lines of symmetry (C
(D) many lines of symmetry
iii) A rectangle has
(A)) two lines of symmetr (A symmetry y
(B) no lines of symmetr symmetry y
(C)) four lines of symmetr (C symmetry y
(D) many lines of symmetr symmetry y
iv) A rhombus has
(A)) no lines os symmetr (A symmetry y
(B) four lines of symmetr symmetry y
(C)) two lines of symmetry (C
(D) six lines of symmetry
v) A scalene triangle has
(A)) no lines of symmetr (A symmetry y
(B) three lines of symmetr symmetry y
(C)) one line of symmetry (C
(D) many lines of symmetry
2. Which of the following have lines of symmetry?
How many lines of symmetry does each have? 3. In the following gures, the mirror line line (i.e. the the line of of symmetry) is given as dotted line. Complete each gure performing reection in the dotted (mirror) line.
168
Geometry 4. Complete the follo following wing table: Shape
Rough gure
Number of lines of symmetry
Equilateral triangle Square Rectangle Isosceles triangle Rhombus
5. Name a triangle which has (i) exactly one line of symmetry. (ii)) exactly three lines of symmetry. (ii sym metry. (iii) no lines of symmetry. 6. Make a list of the capital letters of English alphabets alphabets which (i) have have only one line of symmetry symmetr y about a vertical line.
(ii) have only one line of symmetry about a horizontal line. (iii)) hav (iii havee two lines of symmetry about both horizontal horizontal and vertical line of symmetry.
5.3 Rotational Symmetry Look at the following gures showing the shapes that we get, when we rotate about its centre ‘O’ by an angle of 90
0
or 180
0
Fig. 5.14
Fig. 5.15
169
Chapter 5
Fig. 5.16
In the case ca se of a square, we get exactly the same shape after it is rotated by
90
0
while in the case of a rectangle, we get exactly the same shape after it is rotated by 180° such gures which can be rotated through an angle less than 360° to get the same shape are said to have rotational symmetry.
Angle of Rota Rotatt io ion n The minimum angle a ngle through which the gure has to be rotated to get the original gure is called the angle of rotation and the point about which the gure is rotation is known as centre of rotation. Activity 4:
Take two card board sheets and cut off one equilateral triangle in each sheet such that both the triangles are identi identical. cal. Prepare a circle on a card board and mark the degrees from 0 to 360 degree in the anticlockwise direction. Now palce one one triangle exactly exactly over over the other and put a pin through the centres of of the gures. Rotate the top gure until it matches with the lower gure. You observe that the triangle has been
rotated through an angle 120°. Again rotate the top gure gu re until it matches with the lower gure for the second time. Now you observe that the top of gure has been rotated through an angle 240° from the original position. Rotate the top gure for the third t hird time to match with the lower lower gure. Now the top triangle has reached its original position after a complete rotation of 360° From the above activity you observe that an equilateral triangle has angle of rotation 120°.
170
Geometry
Fig. 5.17
Angle of of rotat r otat io ion n of a hex h exagon agon
Fig. 5.18
In the above Fig. 5.15 to 5.18. We get exactly the same shape of square, rectangle, equilateral triangle and 0
0
0
0
hexagon after it is rotated by 90 , 180 , 120 , 60 respectively. Thus the angle of rotation of (i)
a square is
90
0
0
(ii)
a recta rectangle ngle is 180
(iii)) (iii
an equilateral triangle is 120
(iv)) (iv
a hexagon is
60
0
0
Or der of r otatio otational nal symmetr symmetr y The order of rotational symmetry is the number that tell us how many times a gure looks exactly the same while it takes one complete rotation about the centre. Thus if the angle of rotation of an object is
x
0
It’ss order of rotational symmetr y symmetry It’ symmetr y = In Fig. 5.15 to 5.18. 171
360 x
0
Chapter 5 The order of rotational symmetry of 0
360
(i) a square is
90
0
360
(ii) a recta rectangle ngle is
180
=4
0
0
=2
0
(iii)) an equilateral triangle is (iii
360 =3 120
(iv)) a hexagon is (iv
360
0
60
0
= 6.
Ex ample 5.2
The objects having no line of symmetry can have rotational symmetry symmetr y. Have you you ever made a paper wind wi nd mill? The paper wind wi nd mill in the pictu picture re looks symmet symmetrical. rical. But you you do not nd any line of symmetr y. No folding folding can help you you to have have coincident halves. Howev However er if you rotate it by 90° about about the the centre, the windmill windm ill will look exactly the same. We say the wind mill has a rotationa rotationall symmetry symmetr y.
In a full turn, there are four positions (on rotation through the angles 0
0
0
0
90 , 18 180 270 and 360 ) in which which the wind mill looks exactly the same. Because of
this, we say it has a rotational symmetry of order 4. AcActivity Ac Activity 5:tivity: 5: tivity: 5
As shown in gure cut out a card board or paper triangle. t riangle. Place Place it on a board and x it with a drawi drawing ng pin at one of its vertices. Now rotate the triangle tr iangle about about this vertex, by
90
0
at a time till it comes to
its original position.
172
Geometry You observe that, for every
(i)
90
0
you have the following gures (ii to v).
(ii)
(iii)
(iv)
(v)
The triangle comes back to its original position at position (v) after rotating through
0
360
Thus the angle of rotation of this triangle is
rotational symmetry of this triangle is
360 360
0 0
0
360
and the order of
= 1.
Exercise 5.2
1. Choose the correct answer: i) The angle of rotation of an equilateral triangle is
(A)) (A
60
0
(B) 90
0
(C)) 120 (C
0
(D) 180
0
ii) The order of rotational rotational symmetry of square square is
(A)) (A
2
(B) 4
(C) 6
(D) 1.
iii) The angle of rotation of an object is
(A)) (A
1
(B) 3
72
(C) 4
0
then its order of rotational symmetry is
(D) 5
iv) The angle of rotation of the letter ‘S’ is
(A)) (A
90
0
(B) 180
0
(C)) (C
270
0
0
(D) 360
v) the order of rotational symmetry of the letter ‘V’ is one then its angle of rotation is
(A)) (A
60
0
(B) 90
0
(C)) 180 (C
0
173
0
(D) 360
Chapter 5
2. The following following gures make make a rotation rotation to come to the the new position position about a given centre of rotation. Examine the angle through which the gure is rotated. rotated.
(i)
(ii)
(iii)
(iv)
3. Find the angle of rotation and the order of rotational symmetry symmet ry for the following gures given that the centre of rotation is ‘0’.
(i)
(ii)
(iii)
(iv)
4. A circula circularr wheel has eight spokes.
What is the angle of rotation and the order of rotation?
5.3 Angle Two rays starting from a common point form an angle. In
+ AOB,
O is the vertex,
OA
and
OB
are the two arms. Fig. 5.19
Types of angles (i) Acute angle: An angle whose measure is greater than 0° but less than 90
0
is called an acute angle. a ngle. 0
0
0
0
Example: 15 , 30 , 60 , 75 , In Fig. 5.20
is an acute angle.
+ AOB
=
30
0
Fig. 5.20
174
Geometry (ii) Right angle An angle whose measure is
In Fig. 5.21 + AOB =
90
0
90
0
is called a right angle.
is a right angle. Fig. 5.21
(iii) Obtuse angle An angle whose measure is greater than
90
0
and less
0
than 180 is called an obtuse angle a ngle.. 0
0
0
0
Example: 100 , 110 , 120 , 140 In Fig. 5.22
+
0
AOB = 110 is an obtuse angle. Fig. 5.22
(iv) Straight angle When the arms of an angle, are opposite rays forming a straight line. The angle thus formed is a straight angle and 0
0
whose measure is 180 In Fig. (5.23) + AOB = 180 is a straight
angle.
Fig. 5.23
(v) Refex angle An angle whose measure is more than
180
0
but less
0
than 360 is called a reex re ex angle. In Fig. 5.24 5.24 + AOB = 220°
is a reex angle.
Fig. 5.24
(vi) Complete a ngle In Fig. 5.25 The angle formed by
OP
and OQ is one complete
0
Fig. 5.25
circle, that is 360 .Such an angle is called a complete angle
Related Angles (i)) Complementa (i Complementarr y an gl gles es If the sum of the measures of two angle is
90
0
, then the
two angles are called complementary complementary angles. Here each angle is the complement of the other. 0
The complement of 30 is 0
60 is 30
60
0
and the complement of
0
175
Fig. 5.26
Chapter 5
(ii)) Supplementa (ii Supplementarr y an gl gles es If the sum of the measures of two angle is 180
0
, then the two angles are called supplementary supplementary
angles. Here each angle is the supplement of the other. The supplement of 120
0
is 60
0
and 60° is the supplement of 120
Identify the following pairs
(a) Complement of 85
or supplementar supplementary y 0
0
Fill in the blanks.
of angles are complementar complementary y
(a) 80
Fig. 5.27
0
is____
(b) Complement Complement of 30° is __ _
0
and 10 _____
0
(b) 70 (c) 40
0
(d) 95 (e) 65
(c) Supplement of 60 is __ ____ __
0
and110 _____
0
(d) Supplement of 90 is_____
0
and50 ______
0
0
0
0
and85 _______ 0
and 115 ______
Intersecting lines
Fig. 5.28
Look at the Fig. 5.28. Two Two lines l and l are shown. Both the lines pass through 1
a point P. We say
l1 and l2
2
intersect interse ct at P. P. If two lines have one common point, they are
called intersecting lines. The common point ‘P’ is their point of intersection.
176
Geometry Angless in inter sec Angle secting ting lines When two lines intersect at a point
angles are formed. In Fig. 5.29 the two lines AB and CD intersect at a point ‘O’, + AOD,
+ DOB,
+ BOC
+ COA,
are formed.
Among the four angles two angles are
Fig. 5.29
acute and the other two t wo angles are obtuse.
Fig. 5.30
But in gure 5.30 if the two intersecting lines are perpendicular per pendicular to each other then the four angles are at right r ight angles. angles.
Adjacent angles If two angles have the same vertex and a common arm, then the angles are called adjacent angles. In Fig. 5.31 + BAC and + CAD are adjacent angles (i.e . + x and + y) as they have a common arm AC, a common vertex A and both the angle the common arm
+
+ BAC
and
+ CAD
are on either side of
Fig. 5.31
AC .
ROP and
Look at the following gure
+ QOP
are not adjacent angle. Why?
Open a book looks like the above gure. Is the pair of angles are adjacent angles?
(i)) Adja (i Adjacent cent angles a ngles on a line. lin e. When a ray stands on a straight line two angles are
formed. They are called linear adjacent adjacent angles angles on the line. Fig. 5.32
177
Chapter 5 In Fig. 5.32 the ray OC stands on the line AB.
+ BOC
and
+ COA
are the two
adjacent angles formed on the line AB. Here ‘O’ is called the common vertex,
OC
is
called the common com mon arm. The T he arms ar ms OA and OB lie on the opposite sides of the common arm OC. Two angles are said to be linear adjacent angles on a line if they have a common co mmon ver ver tex tex,, a common common ar m and the other other t wo ar ms ar e on on the opposite opposite si sides des of the common common ar m.
(ii)) The sum of th (ii thee adjacent adja cent an gles on a line lin e is 180° 180°
Fig. 5.33
In Fig. 5.33
+ AOB
Fig.5.34 0
= 180 is a straight angle.
In Fig. 5.34 5.34 The ray OC stands sta nds on the line AB. A B.
+ AOC and + COB
angles. Since
+
AOB is a straight angle whose measure is
+ AOC
+
+
COB = 180
180
are adjacent
0
0
From this we conclude that the sum of the adjacent angles on a line is
180
0
Note 1: A pair of adjacent angles whose non common arms are opposite rays. Note 2: Two adjacent supplementary angles form a straight angle.
Are the angles marked 1 and 2 adjacent? If they are not adjacent, Justify your answer.
178
Geometry
A vegetable chopping board
A pen stand
The chopping blade blade makes a linear li near pair of angles a ngles with the board. The pen makes a linear pair of angles with the stand. Discuss: (i) Can two two adjacent adjacent acute angles form a liner pair? (ii) Can two two adjacent adjacent obtuse angles form a linear pair? (iii) Can two two adjacent adjacent right angles form a linear pair? (iv)) Can an acute and (iv and obtuse obtuse adjacent adjacent angles angles form a linear pair?
(iii)) Angle at a p oint (iii In Fig. 5.35, four angles are formed at the point ‘O’. The sum of the four angles formed is
(i.e)
+1
+
+2
+
+3
+
0
360
+4
O
. 0
= 360
Fig. 5.35
(iv (i v) Ver Ver t ically opposite a ngles If two straight lines AB and CD intersect at a point ‘O’. Then
+ AOC
and
opposite angles and
+
+ BOD
form one pair of vertically
DOA and
+ COB
form another pair
of vertically opposite angles. Fig. 5.36
The following are some real life example for vertically Opposite angles
179
Chapter 5 Activity 6: Draw two lines ‘l’ and ‘m’, intersecting at a point ‘P’ mark + 1, + 2, + 3
+4
and
as in the Fig. 5.37.
Take a trace copy of of the gur guree on a transparent transp arent sheet. Place the copy on the original such that
+
1 matches with its copy,
+ 2,
matches with its copy.. etc...
Fix a pin at the point of intersection of two lines ‘l’ and ‘m’ at P. P. Rotate the copy 0
by 180 . Do the lines coincide again?
Fig. 5.37
You nd that and
+ 4.
+1
and
+3
have interchanged their positions and so have
+2
(This has been done without disturbing the position of the lines).
Thus
+1
=
+3
and
+2
=
+ 4.
From this we conclude that when two lines intersect, the vertically opposite angles are equal. Now No w let us try to prove prove this using using Geometrical idea. idea. Let the lines AB and CD intersect at ‘O’ ‘O’ making angles a ngles + 2, + 3
and
Now
+ 1,
+ 4.
+1
0
= 180 
+2
"
(i)
( Since sum of the adjacent angle on a line +3
0
= 180 
+2
"
180
0
)
0
).
Fig. 5.38
(ii)
( Since sum of the adjacent angle on a line
180
From (i) and (ii) +1
=
+
3 and similarly we prove that
+
Ex ample 5.3 5. 3
In the given gure identify (a) Two pairs of adjacent angles. (b) Two pairs of vertically opposite angles.
180
2=
+
4.
Geometry Solution:
(a) Two pairs of adjacent angles are + EOA, + COE
(i) (ii)
+ COA, + BOC
since OE is common to
+
since OC O C is common to
+ COA
(b) Two pairs of vert vertically ically opposite angles are i) ii)
EOA and
+ COA, + DOB.
x in the given gure. Find the value of x
Solution:
+
+ DCA
= 180
0
+ BCA
(Since
0
= 180 is a straight angle)
45° + x = 180° x = 180° – 45°
= 135° 0
`
x is 135 . The value of x
Ex ample 5.5 x in the given gure. Find the value of x
Solution: + AOD
+
+ DOB
=
(Since
180
0
+ AOB
0
100 + x = 180
0
= 180 is a straight angle)
0
0
x = 180  100
= 80
0
0
0
`
x is 80 . The value of x
Ex ample 5.6 x in the given gure. Find the value of x
Solution: + POR
+
+ ROQ
= 180
( Since
0
+ POQ
0
= 180 is a straight angle)
181
+ BOC
+ BOC, + AOD
Ex ample 5.4
+ BCD
and
+ COE
Chapter 5 x + 2x = 180 3 x = 180 180 x = 3 = 60 `
0
0
0
0
x is The value of x
60
0
Ex ample 5.7 x in the giv Find the value of x g iven en gure.
Solution: + BCD
+
+ DCA
= 180
0
(Since 3 x + x = 180 4 x = 180 180 x = 4
= `
45
+ BCA
0
= 180 is a straights angle)
0
0
0
0
x is The value of x
45
0
Ex ample 5.8 x in the giv Find the value of x g iven en gure.
Solution: + BCD
+
+ DCE
+
+ ECA
= 180
0
(Since 0
0
0
0
0
40 + x + 30 = 18 180 x + 70 = 180
0
+ BCA
x = 180  70 = 110 `
x is 110 The value of x
0
= 180 is a straight angle)
0
0
0
Ex ample 5.9 x in the giv Find the value of x g iven en gure.
Solution: + BCD
+
+ DCE
+
+ ECA
0
= 180 (Since 182
+ BCA
0
= 180 straight angle).
Geometry 0
0
0
0
0
x + 20 + x + x + 40 = 180 3 x + 60 = 180
0
3 x = 180  60 3 x = 120 x = `
0
0
0 120 = 40 3
x is The value of x
40
0
Ex ample 5.10 x in the given gure. Find the value of x
Solution: + BOC
+
+ COA
0
+ + AOD + + DOE + + EOB = 360 (Since angle at a point is
2 x + 4x + 3x + x + 2x = 36 360 12 x = 360 x =
360 12
= 30 `
0
0
0
0
x is 30 The value of x
0
Ex ample 5.11 x the given gure. Find the value of x
Solution: + BOD
(Since
+ + DOE + + EOA = 180
+
0
AOB = 180 is straight angles) 2 x + x + x = 180 4 x = 180 x =
`
0
0
0
180 4
0
0
=
45
x is The value of x
45
0
183
0
360
)
Chapter 5 Exercise: 5.3
1. Choose the correct answer: i) The number of points points common common to two intersecting intersecting line is
(A)) one (A
(B) Two
(C) th three ree
(D) four
ii) The sum of the the adjacent angles on a line is
(A)) (A
90
iii) In the gure
(A)) (A
80
0
+ COA
iv) In the gure 80
(B)) (B 0
+ BOC
90
0
(C) 270
0
(D) 95
0
will be
0
(C) 100
0
will be
0
(C) 100
(A)) (A
(B)) 180 (B
(B)) (B 0
90
0
(D) 120
0
v) In the gure CD is perpendicular to AB. Then the value of + BCE will be
(A)) (A
45
(C) 40
0
(B)) (B
0
(D) 50
35
0
0
2. Name the the adjacent adjacent angles angles in in the the following following gures
3. Identity the vertically opposite angles in the gure.
4. Find (i)
+B
30
(ii)
if +A measures?
0
80
0
(iii) 70° (iv) 60° (v)
45
0
184
0
(D) 360
Geometry 5. In gure AB and CD be the intersecting intersecting lines lines if + DOB
=
35
0
nd the measure of the other angles.
6. Find the value of x x in the following gures.
(i)
(ii)
(iv)
(iii)
(v)
(vi)
7. In the follo following wing gure two lines AB and CD interse intersect ct at x and y. the point O. Find the value of x
8. Two linea linearr adjacent angles on a line are 4 x
and
^3x + 5h
.
Find the value of x. x.
Parallel Lines Look at the table. The top of the table ABCD is a at surf surface. ace. Are you you able to see some some points and line segment on the top top?? Yes. The line segment AB and BC intersects at B. which line segment Fig. 5.39
intersects at A, C and D? Do the line
segment AD and CD intersect? i ntersect? Do the line segment AD and BC intersect? The line segment AB and CD will not meet however they are extended such lines are called parallel lines. AD and BC form one such pair. AB and CD form another pair. If the two lines AB and CD are parallel. We write AB  CD.
185
Chapter 5 The following are the examples of parallel lines
The opposite edges of ruler
The cross bars of this window
l 3
is parallel to
l 3
Two straight lines are said to be parallel to each other if they do not intersect at any point. In the given ﬁgure, the perpendicular per pendicular distance between the two parallel lines is the same everwhere. Fig. 5.40
Transversal A straight line intersects two or more given lines at distinct points is called a transversal t ransversal to the given lines. The given lines may or may not be parallel. Names of angles formed by a transversal. tra nsversal.
The above ﬁgure give an idea of a transversal. You have seen a railway line crossing several lines. (i) Fig. 5.41
(ii)
In Fig. 5.4 5.41 1 (i), a pair of lines AB and CD, are cut by a transversal tra nsversal XY, interse intersecting cting the two lines at points M and N respectively. respectively. The points M and N are called points of intersection. Fig. 5.41 (ii) when a transversal intersects two lines the eight angles marked 1 to 8 have their special names. Let us see what those angles a ngles are
1. Interior angles All the angles which have the line segment MN as one arm in Fig. 5.41 (ii) are known as interior inter ior angles as they lie between the two lines AB and CD. In Fig. 5.4 5.41 1 (ii),
+
3, + 4, + 5, + 6 are interior angles. 186
Geometry 2. Interior Interior alternat e angles angles When a transversal intersects two lines four interior angles are formed. Of the interior angles, the angles that are on opposite sides of the transversal and lie in separate linear pairs are known as interior alternate angles.
+3
and
+ 5, + 4
and
+6
are interior alternate angles in Fig. 5.41 (ii).
3. Exter io iorr angles All the angles which do not have the line segment MN as one arm, are known as exterior angles a ngles.. + 1, + 2, + 7, + 8 are exterior angles in Fig. 5.41 (ii).
4. Exter Exter io iorr alte alterr nate an gl gles es When a transversal intersects two lines four exterior angles are formed. Of the exterior angles, the angles that are on opposite sides of the transversal and lie in separate linear pairs are known as exterior exter ior alternate angles. In Fig. 5.41 (ii), + 1 and
+ 7, + 2
and
+8
are exterior alternate angles.
5. Cor Corrr es espond ponding ing angles angles The pair of angles on one side of the transversal, one of which is an exterior angle while the other is an interior angle but together do not form a linear pair, are known as corresponding cor responding angles. angles. The pairs of corresponding angles in Fig. 5.41 (ii) are +
6, + 3 and
+
+
and
+
5,
+
2 and
7, + 4 and + 8.
Notice that although both and
+1
+
6 is an interior angle while
6 and +7
+
7 lie on the same side of the transversal
is an exterior angle but
+
6 and
+
7 are not
corresponding corr esponding angles as together they form a linear pair. Now we tabulate the angles. a
Interior angles
+
3, + 4,+ 5,+ 6
b
Exterior angles
+
1,+ 2,+ 7,+ 8
c
Pairs of corresponding angles
d
Pairs of alternate interior angles
+3
and + 5 ; + 4 and
+6
e
Pairs of alternate exterior angles
+1
and + 7 ; + 2 and
+8
f
Pairs of interior angles on the same side of the transversal.
+3
and + 6 ; + 4 and
+5
1 and + 5; + 2 and + 6 + 3 and + 7; + 4 and + 8
+
187
Chapter 5 Name the follo following wing angles: angles: a) Any two inter interior ior angles ____ and __ _ _ _ _ _ b) Any two exterior angles ____ and __ _ _ _ _ _ c) A pair of interior angles __ _ _ _ _ _and __ _ _ _ d) A pair of corresponding __ _ _ _ _ _and __ _ _ _ angles.
In Fig. (i) p is a transversal to the lines l and m. In Fig. (ii) the line p is not a transversal, although it cuts two lines l and ‘m’ can you say why?
Properties of parallel lines cut by a transversal Activity Activi ty 7:
Take a sheet of white paper. Draw (in thick colour) two parallel lines ‘l’ and ‘m’. Draw a transversal ‘t ’ to the lines ‘l’ and ‘m’ . Label + 1 and + 2 as shown in Fig 5.42.
Fig. 5.42
Place Pl ace a trace paper over over the gure drawn. Trace the lines ‘l’, ‘m’ and ‘t ’. ’. Slide the trace paper along ‘t ’ until ‘l’ coincides with ‘m’.
You nd that
+1
on the traced gure coincides with
+2
of the original gure.
In fact, you can see all the following results by similar tracing and sliding activity. (i) + 1 = + 2
(ii)
+3
= +4
(iii)
+5
= +6
(iv)
+7
= +8
From this you observe that. When two parallel lines are cut by a transversal, tr ansversal, (a)) each pair of corr (a corresponding esponding angles are equal (b) each pair pair of of alternate angles are equal (c) each pair of interior angles on the same side side of the transversal tra nsversal are 0
supplementary (i.e 180 )
188
Geometry
Draw parallel lines cut by a transversal. Verify the above three statements by actually measuring the t he angles.
Lines l  m, t is a transversal, + x = ?
Lines a  b, c is a transversal, + y = ?
Lines l  m, t is a
l1, l2 be two lines and t is a transversal tra nsversal.. Is + 1 = + 2?
Lines l  m, t is a transversal, + x = ?
transversal, + z = ?
The F  shape stands for corresponding angles.
The Z  shape stands for alternate angles.
189
Chapter 5
Fold a sheet of paper so as to get a pair of parallel lines. Again fold fold the paper across ac ross such that a transversal is obtained. Press the edges of folded folded paper and a nd open it. You You will see a pair p air of parallel lines with the transversal. Measure the angles and verify the properties propert ies of parallel parallel lines when cut by a transversal. trans versal.
Checking for Parallel Lines: Look at the letter z. The horizontal segments are parallel, because the alternate angles are equal.
Ex ample 5.12
In the ﬁgure, ﬁnd +CGH and +BFE .
Solution
In the ﬁgure, AB  CD and EH is a transversal. +FGC
y
= 60° (given) =
+ CGH
= 180° –
+ FGC
(Since
+CGH
and
+FGC
are adjacent angles on a line ) = 180° – 60° = 120°
+ EFA
+
EFA A + EF
= 60° ( Since
+ BFE
= 180° (Since (Since sum of the adjacent angles on a line is 180°) 180°)
+EFA
60° + x = 180° x
= 180° – 60° = 120°
`
x
=
+ BFE
= 120°
y
=
+ CGH
= 120°
190
and +FGC are corresponding angles )
Geometry Ex ample 5.13
In the given gure, nd
+CGF
and +DGF .
Solution
In the gure AB  CD and EH is a transversal. + GFB
= 70°
(given)
+ FGC
= a = 70° (Since alternate interior angles +CGF
+ CGF
+ + DGF = 180°
+GFB
and
are equal)
(Since (S ince sum of the adjacent angle on a line is 180°)
a + b = 180°
70 + b = 180° b = 180° – 70° 70°
= 110° + CGF
= a = 70°
+ DGF
= b = 110°
Ex ample 5.14
In the given gure,
+ BFE
= 100°
and + CGF = 80°. Find i) iii)
+ EFA,
+ GFB,
iv)
ii) + DGF, + AFG,
v) + HGD.
Solution + BFE
= 100° and
EFA =
+ 80°
+ CGF
= 80° (given)
(Corresponding angles)
i)
+
ii)
+ DGF
= 100°
(Since (S ince corre corresponding sponding angles are equal)
iii)
+ GFB
= 80°
(Since (S ince alternate interior angles are equal)
iv)) iv
+ AFG
= 100°
(Since (S ince corre corresponding sponding angles + AFG
v)
+ HGD
= 80°
+ CGH
and
are equal)
(Since (S ince corre corresponding sponding angles are equal)
191
Chapter 5 Ex ample 5.15
In the ﬁgure, AB  CD,
+ AFG
= 120° Find
(i) + DGF (ii)
+ GFB
(iii)
+ CGF
Solution
In the ﬁgure, AB  CD and EH is a transversal (i)
+ AFG
= 120°
+ AFG
=
+ DGF
= 120°
are equal)
+
+ GFB
= 180°
(Since sum of the adjacent angle
120° +
+ GFB
= 180°
on a line is 180°)
+ GFB
= 180° – 120°
`
(ii)
+ AFG
(Given)
+ DGF
= 120° (Since alter alternate nate inter interior ior angles
= 60° (iii)
+ AFG
+
+ CGF
= 180°
120° +
+ CGF
= 180°
(Since sum of the adjacent angles on a line is 180°)
+ CGF
= 180° – 120° = 60°
Ex ample 5.16
Find the measure of x x in the ﬁgure, given
l
 m.
Solution
In the ﬁgure, l  m +3
3 x +
x
=
x
(Since (S ince alternate interior angles are equal)
= 180° (Since sum of the adjacent angles on a line is 180°)
4 x = 180° x
=
180 4
0
= 45°
192
Geometry Exercise 5.4
1. Choose the correct answer i) If a transversal transversal intersect two lines, the number of angles formed are (A)) 4 (A
(B) 6
(C) 8
(D) 12
ii) If a transversal intersect any two two lines lines the two lines lines (A) are parallel (C) may or may not be parallel
(B) are not parallel (D) are perpendicular
iii) When two parallel lines are cut by by a transversal, transversal, the sum of the the interior angles on the same side of the transversal is (A)) 90° (A
(B) 180°
(C) 270°
(D) 360°
iv) In the given gure + BQR
and
+ QRC
are a pair of
(A) vertically apposite angles (B) exterior angles (C) alternate interior angles (D) corresponding angles
v) In the given gure + SRD = 110° then the value of + BQP will be (A) 110°
(B) 100°
(C) 80°
(D) 70°
2. In the given gure, gure, state the property property that is used in each of the following statement. (i) If l  m then + 1=+ 5. (ii) If + 4 = + 6 then l  m. (iii) If + 4 +
+5
= 180° then l  m.
3. Name the required angles in the gure. (i) The angle vertically opposite to (ii) The angle alternate to
+AMN
+ CNQ
(iii) The angle corresponding to
+
BMP
(iv) The angle corresponding to
+
BMN
4. In the given gure identify (i) Pairs of corresponding angles (ii) Pairs of alternate interior angles. (iii) Pairs of interior angles on the same side of the transversal (iv) Vertically opposite angles.
193
Chapter 5 5. Given l  m, ﬁnd the measure of x in the following ﬁgures
6. Given l  m and +1 = 70°, ﬁnd the measure of +2, +3 , +4, +5, +6, +7 and +8 .
7. In the given ﬁgures below, decide whether l  m? Give reasons.
8. Given l  m, ﬁnd the measure of + 1 and + 2 in the ﬁgure shown.
Triangle: Revision A triangle is a closed plane
ﬁgure
made of three line
segments. In Fig. 5.43 the line segments AB, BC and CA form a closed D ABC.
ﬁgure.
This is a triangle and is denoted by
This triangle may be named as
D ABC
or
D BCA
D CAB.
or
Fig. 5.43
The line segments forming a triangle are the three sides of the triangle. In Fig.5.43 AB , BC and CA are the three th ree sides of the triangle. 194
Geometry The point where any two of the three line segments of a triangle intersect is called the vertex of the triangle. In Fig. 5.43 A,B and C are the three vertices of the D ABC.
When two line segments intersect, they form an angle at that point. In the triangle in Fig. 5.43
AB
and
BC
intersect at B and form an angle at that vertex. This
angle at B is read as angle B or + B or + ABC. Thus a triangle has three angles angles + A, +B
and + C. In Fig. 5.4 5.43 3
D ABC
Sides
:
AB,BC,CA
Angles
:
BCA A + CAB, + ABC, + BC
Ver ertices tices :
has
A, B, C
The side opposite to the vertices A, B, C are BC, AC and AB respectively. The angle opposite to the side BC, CA and AB is
+ A, + B
and
r espectively ively.. + C respect
A triangle is a closed closed gure made of three line segments. segments. It has three vertices, three sides and three angles a ngles..
Types of Triangles Tr iangles Based on sides A triangle is said to be
Equilateral, when all its sides are equal. Isosceles, when two of its sides are equal. Scalene, when its sides are all unequal.
Based on angle a ngless A triangle is said to be
Right angled, when one of its angle is a right angle and the other two angles are acute. Obtuse  angled, when one of its angle is obtuse and the other two angles are acute. Acute  angled, when all the three th ree of its angles are acute. a cute. The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. 195
Chapter 5 Angle sum property of a triangle: tr iangle: Activity 8
Draw any triangle ABC on a sheet of paper and mark the angles 1, 2 and 3 on both sides of the paper
Is it possible to form a triangle whose sides are 7cm, 5cm and a nd 13cm?
as shown in Fig. 5.44 (i).
Fig. 5.44
Cut the triangle tria ngle ABC. Fol Fold d the vertex A to touch touch the side BC as shown shown in the Fig. 5.44 (ii) Fold the vertices B and C to get a rectangle as shown in the Fig. 5.44 (iii) Now you see that
+ 1, + 2
and
+3
make a straight line.
From this you observe that +1 +A
+ +2 +
+
+B
+
+3
+
= 180°
C = 180°
The sum of the three angles of a triangle is 180° Activity 9
Draw a triangle tr iangle.. Cut on the three angles. Re arrange ar range them t hem as shown in Fig. 5.45 5.45 (ii). (ii). You You observe that the three angles now constitute one angle. This angle is a straight angle and so Fig. 5.45
has measure 180° The sum of the three angles of a triangle is 180° Think it.
1. Can you have have a triangle with the three t hree angles less than 60°? 2. Can Can you have a triangle with two right angles?
196
Geometry Exteriorr angle of a tr iangl Exterio ianglee and its propert y Activity 10
Fig. 5.46
Draw a triangle ABC and produce one of its sides, say BC as shown in Fig. 5.46 (i) observe the angles ACD formed at the point C. This angle lies in the exterior of T ABC formed at vertex C. + BCA is an adjacent
angle to + ACD ACD.. The remaining remaini ng two angles of the triangle tria ngle
namely + A nd + B are called the two interior i nterior opposite opposite angles.
Now No w cut out (or (or make trace copies of)
+A
and
+B
and place them adjacent to
each other as shown in Fig. 5.46 (ii) You observe that these two pieces together entirely cover
+
ACD.
From this we conclude that the exterior angle of a triangle is equal to the sum of the two interior opposite angles. The relation between an exterior angle and its two interior angles is referred to as the exterior angle property of a triangle tr iangle.. Draw a triangle tr iangle ABC and produce one of its sides BC as shown in Fig. 5.46 (i) + ACD formed at the point point C. Now take a protractor and measure + ACD, + A and + B. Find the sum + A + + B. and compare it with the measure of + ACD. Do you observe that + ACD = + A + B? Ex ample 5.17 5.17 x. In the given gure nd the value of of x.
Solution + CAB
+
+ ABC
+ + BCA = 180° 40° + x + x = 180° 40° + 2 x = 180°
(Since sum of the three
angles of a triangle is 180°)
2 x = 180° – 40° 2 x = 140° x =
140° 2
= 70°
x = 70°. The value of x 197
Chapter 5 Ex ample 5.18
Two angles of of a triangle tria ngle are 40° and 60°. 60°. Find the third angle. Solution + RPQ
+
+ PQR x
++ QRP = 180°
(Since sum of the thre threee angles of a
+ 40° + 60° = 180° 180° x
triangle tr iangle is 180°)
+ 100° = 180° x
= 180° – 100° = 80°
`
The third angle
x
= 80°
Ex ample 5.19
In the given ﬁgure, ﬁnd the measure of + A.
Solution + CAB
+
+ ABC
+
+ BCA
= 180°
(Since sum of the thre threee angles of a
2 x + 120° + x = 180°
triangle tr iangle is 180°)
3 x + 120° = 180° 3 x = 180° – 120° 3 x = 60° x
=
60° 3
= 20° `
+A
= 2 x = 2 × 20° = 40°
Ex ample 5.20
In the given ﬁgure. Find the value of x. Solution
In the ﬁgure exterior angle =
+ ABD
= 110°.
Sum of the two interior opposite angle =
+ BCA
+
+
CAB
= x + 50° x
+ 50° = 110° x
= 110° – 50°
(Since the sum of the two inter interior ior opposite angle is equal to the exterior angle)
= 60° `
The value of x is 60°.
198
Geometry Ex ample 5.21 x and y. In the given gure nd the values of of x
Solution
In the give gure, Exterior angle =
+
DCA = 130°
50° + x = 13 130° 0°
(Since (S ince sum of the two inter interior ior opposite
x = 13 130° 0° – 50° 50° angle is equal to the exterior angle)
= 80° In
D ABC, +A
+ + B + + C = 180° (S (Since ince sum of thre threee angles of of a tr triangle iangle is 180°) 50° + x + y = 180°
50° + 80° + y = 180° 130° + y = 180° y = 180° – 130°
= 50° `
x = 80° and y = 50°. The values of x
Aliter: + ACB
+
+
DCA = 180° (S (Since ince sum of the adjacent adjacent angles on a line is 180°)
y + 130° = 180° y = 180° – 130°
= 50° In
D
+A
ABC, + + B + + C = 180° (S (Since ince sum of the thre threee angles of a tr triangle iangle is 180°) 50° + x + y = 180° 50° + x + 50° = 180° 100° + x = 180° x = 180° – 100°
= 80°
199
Chapter 5 Ex ample 5.22
Three angles a ngles of of a triangle tr iangle are 3 x + 5° 5°, x + 20° 20 °, x + 25°. 25°. Find the measure mea sure of each ea ch angle. Solution
Sum of the thre threee angles of a tr triangle iangle = 180° 3x + 5° 5° + x + 20° +
x
+ 25° = 180° 180°
5x + 50° = 180° 180° 5x = 180° – 50° 5x = 130° x
= 130° 5
= 26° 3x + 5° = (3 × 26°) 26°) + 5° = 78° + 5° 5° = 83°
`
x
+ 20° = 26° + 20° = 46°
x
+ 25° = 26° + 25° = 51 51°°
The three angles of a triangle are 83°, 46° and 51°. Exercise 5.5
1. Choose the correct answer. i) The sum sum of the three three angles angles of a triangle triangle is is (A)) 90° (A
(B) 180°
(C) 270°
(D) 360°
ii) In a triangle, all the three angles are equal, then the measure of each angle is (A)) 30 (A
0
(B) 45
0
(C)) 60 (C
0
(D) 90
0
iii) Which of the following can be be angles angles of a triangle? triangle? 0
0
(A)) 50 , 30 , 105 (A
0
0
0
(B) 36 , 44 , 90
0
0
0
(C) 70 , 30 , 80
0
0
iv) Two angles of a triangle are 40° and 60°, then the third angle is (A)) 20 (A v) In
0
(B) 40
0
(C)) 60 (C
0
(D) 80 0
T ABC,
BC is produced to D and +ABC = 50 , 0 +ACD = 105 , then + BAC will be equal to
(A) 75°
(B) 15°
(C) 40°
(D) 55 55°°
200
0
0
(D) 45 , 45 80 80 0
Geometry 2. State which of the following are triangles. (i)
+A = 25
(ii)
+P
(iii)
0
= 90
+X
=
0
40
0
0
+B
= 35
+Q
= 30
+Y
= 70
0
0
+C
= 120
+
R=
50
+
Z=
80
0
0
0
3. Two angles of a triangle is is given, nd the third angle. 0
0
(i) 75 , 45
0
(ii) 80 , 30
0
0
(iii) 40 , 90
0
0
0
(iv) 45 , 85
4. Find the value of the unknown x in the following diagrams:
5. Find the values of the unknown x and y in the following diagrams:
6. Three angles of a triangle are x + 5°, x + 10° and x + 15° nd x.
201
Chapter 5
1. Symmetr y refers to the th e exact exact match in shape shape and siz size e bet bet ween two halves of an object.
2. When Whena alin lined edivi ivides desag agiv iven engu gurei reinto ntotwo twoequ equal alhal halves vessu sucht chthat hatth thel elefta eftand nd righthalvesmatchesexactlythenwesaythatthegureissymmetricalabout the line. This line is called the line of symmetry or axis of symmetry.
3. Eac Eachr hreg egul ular arpol polyg ygon onha hasa sasm sman anyl ylin ines esof ofsy symm mmetry etryas asit ith has assi sides des.. 4. So Some meo obj bject ects san and dg gur ures esh ha ave ven no oli line nes sof ofs sym ymme metry try.. 5. Fig Figures uresw which hichca canb nber erotat otated edthro through ugha ana nangl nglel eless esstha than n360 360°t °to oget getthe thesame same shape are said to have rotational symmetry.
6. The Theord ordero erofro frotat tatio iona nalsym lsymmet metryis ryisthe thenum number berth thatt attel ellus lushow howma many nytim times es agurelooksexactlythesamewhileittakesonecompleterotationaboutthe centre. 7. The object objects s having no line of symmetry symmetr y can can have have rot rotational ational symmet symmetry. ry. 8. I f t wo angles angles hav have e the same same vert vertex ex and and a common arm, arm, then t hen the angles angles are called adjacent angles.
9. Th The esu sum mof oft the hea adj djac acen ent tan angl gles eso on na ali line nei is s1 180 80°°. 10. When Whentwo twoli linesi nesinte nterset, rset,the theve vertica rtically llyop opposi positea teangl nglesa esareeq reequal ual.. 11. An Ang gle lea at ta apo poiint nti is s36 360° 0°.. 12. Two st st raight lines are are said said to be parallel parallel to each each other if t hey do not int ersec ersectt at any point. 13. A straight line lin e intersects two tw o or more lines at distinct point points s is called called a transversal to the given line.
14. Wh When entwo twopa para rallllel ellilines nesar arec ecut utby bya atr tran ansv sversa ersal, l, (a) )ea each chp pai air rof ofc cor orre resp spon ond ding nga ang ngle les sa are ree eq qua uall. (b) )e each pair ro of fa alter ern nate ea angles esa arre eeq equ ual. (c) each pair of interior angles on the same side of the transversal are
supplementary. 15. Th Thes esum umof ofth thet ethr hree eean angl gles esof ofa atr tria iang ngle leis is1 180° 80°.. 16. In Inatr atrian iangle glean anexteri exteriora orangl ngleis eisequ equalt altoth othesu esumof mofth thetwo etwoin inter terior ioroppo opposit site e angles.
202
P ra rac cti tic cal Geome Ge ome try
6.1 6. 1 Intr oduction This chapter helps the students to understand and conrm the concepts they
have learnt already in theoretical geometry. This also helps them to acquire some basic knowledge knowledge in geometry geometry which they are going going to prove prove in their later classes. No doubt, all the students will do the constructions constr uctions actively actively and learn lear n the concepts easily. easily. In the previous class we have learnt to draw a line segment, the parallel lines, the perpendicular lines and also how to construct an a n angle. Here we are going to learn about the construction of perpendicular bisector of a line segment, angle bisector, some angles using scale and compass and the constr uction of triangles.
Review To recall r ecall the concept concept of angles, angles, par allel lines and p er pend pendic icular ular lines from the given gure.
We shall identify the points, the line segments, the angles, the parallel lines and the perpendicular lines from the gures given below in the table.
Figures
1
Points identied
Lines identied
A, B, C and D
AB, BC, CD, AD, and BD
Angles identied
1234
203
Parallel lines
+ BAD ( + A) + DCB ( + C) + DBA + CBD
AB  DC BC  AD
Perpendicular line liness
AB = AB = BC = CD =
AD BC CD AD
Chapter 6
No.
F igu r es
Points Lines iden identied tied
Angles identied
Parallel lines
2
3
6.2 Perpendicular bisector of the given line segment (i) Act Activity ivity : Pap er folding folding
•
Draw a line segment AB on on a sheet of paper.
X
•
Fold Fo ld the paper so that the end point point B lies on A. Make a crease XY on the paper. Y X
•
Unfold the paper. Mark the point O where the line of crease XY intersects the line AB. A B.
O
Y
204
Perpendicular lines
Practical Geometry •
By actual measurement we can see that OA OA = OB and the line of of crease XY is perpendicular to the line AB.
The line of crease XY is the t he perpendicular bisector of the line AB. The perpendicular per pendicular bisector bisector of a line li ne segment is a perpendicular line drawn at its midpoint. midpoint.
(ii) To construct a perpendicular bisector to a given line segment. Step 1 : Draw a line segment
AB
of
the
given
measurement.
Step 2 : With ‘A’ as centre draw
arcs of radius more than half of AB, above and below belo w the line AB.
Step 3 : With ‘B’ as centre and
with the same radius draw two arcs. These arcs cut the previous arcs at P and Q.
205
Chapter 6
Step 4 : Join PQ. Let PQ intersect
AB at ‘O’.
PQ is a perpendicular
bisectorr of AB. bisecto
Mark any point on the perpendicular bisector PQ. Verify Verify that it is equidistant from both A and B.
The perpendicular bisector of a line segment is the axis of symmetry for the line segment.
Ex ample 6.1 6 .1
Draw a perpendicular bisector to the
line segment AB = 8 cm. Solution
Step 1 : Draw the line segment
AB = 8cm. Step 2 : With ‘A’ as centre draw
arcs of radius more than half of AB above and below belo w the line AB. Step 3 : With ‘B’ as centre draw
the arcs of same radius to cut the previous arcs at X
and Y. 206
Can there be more than one perpendicularr bisector for the perpendicula given line segment?
Practical Geometry Step 4 : Join XY to intersect the line AB at O.
XY is the perpendicular bisector of AB. 1. With PQ = 6.5 cm as diameter draw a circle. 2. Draw a line segment of of length 12 cm. Using compass divide it into four equal parts. par ts. Verify Verify it by actual measurement. 3. Draw a perpendicular bisector to a given line segment AC. Let the bisector intersect the line at ‘O’. Mark the points B and D on the bisector at equal distances from O. Join the points A, B, C and D in order. Verify whether all lines joined are of equal length. Think!
In the above construction mark the points B and D on the bisector, bisecto r, such that OA OA = OB = OC = OD. OD. Join the points points A, B, C and D in i n order. Then 1. Do the lines joined are of equal length? 2. Do the angles a ngles at the vertices are right angles? angles? 3. Can you identify the gure?
6.3 Angle Bisector (ii) Activity : Pap er folding folding
•
Take a sheet of paper and mark a point O on it. With O as initial point draw two rays OA and OB to make
+ AOB.
C
•
Fold Fol d the sheet sheet through ‘O ‘O’’ such that the rays OA and OB coincide with each other and a nd make a crease on the paper.
207
Chapter 6 •
Let OC be the line of crease on the paper after unfold. By actual measurement, + BOC
•
+ AOC
and
are equal.
So the line of crease OC divides the given angle into two equal parts.
•
This line of crease is the line of symmetr symmetry y for + AOB.
•
This line of symmetry for + AOB is called the angle bisector. The angle bisector of a given angle is the line of symmetr y which divides divides the angle a ngle into two equal parts. par ts.
(ii)) To (ii To constr constr uct a n angle a ngle bisector bisector of th thee given given an gle using scale and compass
Step 1 : Construct an angle of given given
measure at O.
Step 2 : With ‘O ’ as centre draw an
arc of any radius to cut the rays of the angle at A and B.
Step 3 : With ‘A’ as centre draw an
arc of radius more than half of AB, in the interior of the given angle.
208
Practical Geometry
Step 4 : With ‘B’ as centre draw an
arc of same radius to cut the previous previo us arc at ‘C’ ‘C’.
Step 5 : Join OC.
OC is the angle bisector of the given angle.
Mark any point on the angle bisector OC. Verify that it is equidistant from the rays OA and OB.
Ex ample 6.2 6 .2
Construct
+ AOB
= 80° and draw its angle bisector.
Solution
Step 1 : Construct + AOB = 80° angle a ngle at
the point ‘O’ using protractor. Step 2 : With ‘O’ as centre draw an arc
of any radius to cut the rays r ays OA OA and OB at the points X and Y respectively. Step 3 : With ‘X’ as centre draw an arc
of radius more than half of XY in the interior of the angle.
209
C
Chapter 6 Step 4 : With ‘Y’ ‘Y’ as centre draw d raw an arc of the same radius ra dius to cut the previous
arc at C. Join OC. OC is the angle bisector of the given angle 80°.
Draw an angle of measure 120° and divide into four equal parts. Exercise 6. 6.1 1
1. Draw the line segment AB = 7cm 7cm and construct construct its perpendicular bisector bisector.. 2. Draw a line line segment XY XY = 8.5 cm and nd its axis of symmetry. symmetry. 3. Draw a perpendicular bisector of the line segment AB = 10 cm. 4. Draw an angle measuring 70° and construct its bisector. 5. Draw an angle measuring 110° and construct its bisector.
6. Construct a right angle and bisect it using scale and compass.
1. Draw a circle with centre ‘C’ and radius 4 cm. Draw any chord AB. AB. Construct perpendicular perpendicular bisector to AB and examine whether it passes through the centre of the circle. 2. Draw perpendicular bisectors to any two t wo chords chords of equal length in a circle. (i) Where do they meet? (ii) Verify whether the chords are at a same distance from the centre. 3. Plo Plott three points not on a straight line. Find a point equidistant from them. Hint: Join all the points in order. You get a triangle. Draw perpendicular Hint: Join bisectors to each side. They meet at a point which is equidistant from the points you have plotted. plotted. This point is called circumcentre.
6.4 To construct angles 60°, 30°, 120°, 90° using scale and compass. (i)) Constr uct (i uctio ion n of 60° an gle Step 1 : Draw a line ‘l’ and mark a
point ‘O’ ‘O’ on on it.
210
Practical Geometry Step 2 : With ‘O’ as centre draw an arc of any radius to cut the line at A. Step 3 : With the same radius and A as centre draw an arc to cut the previous arc at B. Step 4 : Join OB. + AOB
= 60°.
Draw a circle of any radius with centre
‘O’. Take any point ‘A’ on the circumference. With ‘A’ as centre and OA as radius draw an arc to cut the circle at ‘B’. Again with ‘B’ as centre draw the arc of same radius to cut the circle at ‘C’. Proceed so on. The nal arc will pass through the the point point ‘A’. Join Join all such such points A, B, C, D, E and F in order. ABCDEF is a regular Hexagon. From the above gure we came to know (i) The circumference of the circle is divided into six equal arc length length subtending 60° each at the centre. In any circle a chord of length equal to its radius subtends 60° angle at the centre. (ii) Total angle measur measuring ing around a point is 360° 360°.. (iii)
It consists of six equilater equilateral al tria triangles. ngles.
(ii)) Constr uct (ii uctio ion n of 30° an gle First you construct 60° angle and then bisect it to get 30° angle. Step 1 : Construct 60° (as shown in the above construction (i))
Step 2 : With ‘A’ as centre, draw
an arc of radius more than half of AB in the interior inter ior of of + AOB.
211
Chapter 6 Step 3 : With the same radius and with B as centre draw an arc to cut the
previous previo us one at C. Join OC. + AOC
is 30°.
How will you construct 15° angle.
(iii)) Constr uct (iii uctio ion n of 120° 120° angle an gle Step 1 : Mark a point ‘O’ on a line ‘l’.
Step 2 : With ‘O’ as centre draw
an arc of any radius to cut the line l at A.
Step 3 : With same radius and with ‘A’ as centre draw another arc to cut the
previous previo us arc at ‘B’ ‘B’.
Step 4 : With ‘B’ as centre draw
another arc of same radius to cut the rst arc at ‘C’. Step 5 : Join OC. + AOC
is 120°.
212
Practical Geometry (iv (i v) Constr uct uctio ion n of 90° angle To construct 90° angle, we are going to bisect the straight angle 180°.
Step 1 : Mark a point ‘O’ on a
straight line ‘l’. Step 2 : With ‘O’ as centre draw
arcs of any radius to cut the line l at A and B. Now No w + AOB = 180°.
Step 3 : With A and B as centres and with the radius
more than half of AB draw arcs above AB to instersect each other at
‘C’.
Step 4 : Join OC. + AOC
= 90°.
1. Construct an angle of measure 60° and nd the angle bisector of its
To construct a perpendicular per pendicular
for a given line at any point
complementary angle.
on it, you can adopt this
2. Trisect the right angle.
method for the setsquare
3. Construct the angles of following measures: 22½°, 75°, 105°, 135°, 150°
method, as an alternate.
Exercise 6.2
1. Construct the angles angles of following measures with ruler and compass. (i) 60°
(ii) 30°
(iii) 120°
213
(iv) 90°
Chapter 6 6.5 Co Constr nstr uction of of tr iangle iangless In the previous class, we have learnt the various types of triangles on the basis of using their sid sides es and angles angles.. No Now w let us recall the differen differentt types of triangles and some properties of triangle. Classication of triangles No.
1 S E D I S s i s a b 2 S e E h t L n G o
N A I R 3 T F O N O I T A C 4 I F I S S S E A L L G C N A s 5 i s a b e h t n o
6
Na m e of Tr ia n gle
F igu r e
Equilateral triangle
Not e
Three sides are equal
Isosceles triangle
Any two sides are equal
Scalene triangle
Sides are unequal
Acute angled triangle
All the three angles are acute (less than 90°)
Obtuse angled triangle
Any one of the angles is obtuse (more than 90°)
Right angled triangle
Any one of the angles is right angle (90°)
Some So me pr oper operties ties of of tr iangle
1.
The sum of the lengths lengths of any two sides sides of a tr triangle iangle is greater than the third side.
2.
The sum of all the thre threee angles of a tr triangle iangle is 180° 80°..
214
Practical Geometry To construct a triangle we need three measurements in which at least the length of one side must be given. Let us construct the following types of triangles with the given measurements. (i) Thre Threee sides (SS (SSS). S). (ii) Two sides sides and included angle between them (SAS). (SAS). (iii) Two angles and included side side between them (ASA) (ASA)..
(i)) To (i To constr constr uct a tr iangle when when thr t hr ee sides sides are given given (SSS Cr iterion) Ex ample 6.3 6 .3
Construct a triangle t riangle ABC given given that AB = 4cm, BC = 6 cm and AC = 5 cm. Solution
Given measurements
Rough Diagram
AB = 4cm BC = 6 cm AC = 5 cm.
Steps for construction
Step 1 : Draw a line segment BC = 6cm Step 2 : With ‘B’ as centre, draw an arc of radius 4 cm above the line BC. Step 3 : With ‘C’ as centre, draw an arc of 5 cm to intersect the previous arc at ‘A’ Step 4 : Join AB and AC. Now ABC is the required triangle.
215
Chapter 6
By using protector measure all the angles of a triangle. Find the sum of all the three angles of a triangle.
1. A student attempted to draw draw a triangle with given measurements PQ = 2cm, QR = 6cm, PR = 3 cm. (as in the rough gure). First he drew QR = 6cm. Then he drew an arc of 2cm with P as centre and he drew an arc of radius 3 cm with R as centre. They could not intersect each to get P.
(i) What is the reason? (ii) What is the triangle property in connection with this?
The sum of any two sides of a triangle is always greater than the third side.
Draw the bisectors of the three angles of a triangle. Check whether all of them pass through a same point. This point is incentre.
(ii)) To constr (ii constr uct a t r iangle when when Two Two sides sides an and d an an angle gle included bet we ween en them ar e given. given. (SA (SAS S Cr iterion) Ex ample 6.4 6 .4
Construct a triangle t riangle PQR given given that PQ = 4 cm, QR = 6.5 cm and a nd + PQR = 60°. Solution
Given measurements PQ
=
QR = + PQR
=
4 cm 6.5 cm 60°
216
Practical Geometry
Steps for construction
Step 1 : Draw the line segment QR = 6.5 cm. Step 2 : At Q, draw a line QX making an angle of 600 with QR. Step 3 : With Q as centre, draw an arc of radius 4 cm to cut the line (QX)
at P. Step 4 : Join PR.
PQR is the required triangle tr iangle..
Construct a triangle with the given measurements XY = 6cm, YZ = 6cm and a nd + XYZ = 70°. Measure the angles of the triangle opposite to the equal sides. What do you observe?
(iii) (i ii) To constr constr uct a tr iangle when when t wo of its its an gle gless and a si side de included between them are given. (ASA criterion) Ex ample 6.5 6 .5 Construct a triangle XYZ given that XY = 6 cm, + XYZ = 100°. Examine whether the third angle measures 50°. Solution Given measurements
XY = 6 cm + ZXY
= 30°
+ XYZ
= 100°
217
+ ZXY
= 30° and
Chapter 6
Step 1 : Draw the line segment XY = 6cm. Step 2 : At X, draw a ray XP making an angle of 30° with XY. Step 3 : At Y, draw another ray YQ making an angle of 100° with XY. The
rays XP and YQ intersect at Z. Step 4 : The third angle measures 50° i.e
+Z
= 50°.
Construct a triangle t riangle PQR given given that PQ = 7 cm, +Q
= 70°, + R = 40°.
Hint:: Use the Angle Sum Property of a triangle Hint tr iangle.. Exercise : 6.3
I. Construct the triangles for the following given measurements. 1. Construct
3 PQR,
given that PQ = 6cm, QR = 7cm, PR = 5cm.
2. Construct an equilateral triangle with with the side 7cm. Using protector protector measure each angle of the triangle. Are Are they equal? 3. Draw a triangle triangle DEF such that DE = 4.5cm, 4.5cm, EF = 5.5cm and DF = 4.5cm. 4.5cm. Can you indentify the type of the triangle? Write the name of it. II. Construct the triangles for the following given measurements. 4. Construct
3 XYZ,
5. Construct
3 PQR
given that YZ = 7cm, ZX = 5cm,
when PQ = 6cm, PR = 9cm and
+Z
+P
= 50°.
= 100°.
6. Construct 3ABC given that AB = 6 cm, BC = 8 cm c m and length of AC.
+B
= 90° measure
III. Construct the triangles for the following given measurements. 7. Construct
3 XYZ,
8. Construct
3ABC
9. Construct
3 LMN,
when X = 50°, Y = 70° and XY = 5cm.
when A = 120°, B = 30° and AB = 7cm. given that
+L
= 40°,
write the length of sides opposite to the of Triangle is this? 218
+M
+L
= 40° and LM = 6cm. Measure and
and
+ M.
Are they equal? What type
DATA DA TA H AND NDLIN LING G Introduction Data Handling is a part of statistics. The word statistics is derived from the Latin word “ Status”. Like Mathematics, Statistics is also a science of numbers numbers.. The numbers referred to here are data d ata expressed in numerical form like, (i)
Marks of students in a class
(ii) Weight of children of par particular ticular age in a village (iii) The amount of rainfall in i n a region over over a period of years years..
Statistics deals with the methods of collection, classication, analysis and interpretation of such data.
Any collection of information in the form of numerical gures giving the required information is called data.
Raw data The marks obtained in Mathematics test by the students st udents of a class is a collection collection of observations gathered initially. The information which is collected initially and presented randomly is called a raw data.
The raw data is an unprocessed and unclassied unclassied data.
Gr ouped data Some times the collected raw data may be huge in number and it gives us no information as such. Whenever the data is large, we have to group them meaningfully and then analyse a nalyse.. The data which is arranged ar ranged in groups or classes is called a grouped data.
Collection of data The initial step of of investigation is the collection of of data. The collected data must be relevant to the need.
219
Chapter 7 Pr im imar ar y data data For example, Mr. Vinoth, the class teacher of standard VII plans to take his students for an excursion. He asks the students to give their choice for (i)
particular par ticular location they would like to go
Find the relevant data for the students from tribal villages are good visual learners.
(ii) the game they would like to play (iii) the food they would like to have have on their trip tr ip For all these, he is getting getting the information directly directly from the students. This type of collection collection of of data is known as primary pri mary data.
7.1 Collectio Collection n a nd Or ganizing of Continues Data Secondary data Mr. Vinoth, the class teacher of standard VII is coll collecting ecting the information i nformation about about weather for for their trip. He may collects collects the information information from the internet, news news papers, magazines, television television and and other sources. These external sources are called secondar secondary y data.
Variable As for as statistics is concerned the word variable means by measurable measurable quantity which takes any numerical values values within certain cer tain limits. Few etxamples are (i) age, (ii) income, (iii) height and (iv) weight.
Frequency Suppose we measure the height of students in a school. It is possible that a particular par ticular value of height say 140 140 cm gets repeated. rep eated. We then count the number of times the value occurs. This number is called the frequency f requency of 140 140 cm. The number of times a particular par ticular value repeats itself is called its frequency f requency..
Range The difference between the highest value and the lowest value of a particular data is called the range. Ex ample 7.1
Let the heights (in cm) of 20 students in a class be as follows. 120, 122, 127, 112, 129, 118, 130, 132, 120, 115 124, 128, 120, 134, 126, 110, 132, 121, 127, 118. Here the least value is 110 cm and the highest value is 134 cm. Range = Highest value  Lowest value = 134 – 110 = 24 220
Data Handling Class and C lass Int er erv val The above example we take 5 classes say 110  115, 115, 120, 120  125, 125  130, 130  135 and each class is known as class interval. The class interval must be of equal size. The number of classes is neither too big nor too small. i.e The optimum number of classes is between 5 and 10.
Class limits In class 110  115, 110 is called the lower limit of the class and 115 is the upper limit.
Width (or size) size) of of the class class inter inter val: The difference between the upper and lower limit is called the width of the class interval. inter val. In the above example the width of the class inter val is 115 115  110 = 5. By increasing the class interval, inter val, we can reduce the number of classes. There are two types of class intervals. They are (i) inclusive form and (ii) Exclusive form.
(i) In Inclusiv clusivee fo forr m In this form, the lower lower limit as well as upper limit will be included in that class
interval. For example in the rst class interval 110  114 the heights 110 as well as 114 are included. In the second class interval 115  119 both the heights 115 and 119 are included and so on.
(ii) Exclusive form: In the above example 7.1, in the rst class interval 110  115, 110 cm is included and 115 115 cm is excluded. excluded. In the second se cond class interval inte rval 115 115 is included and 120 is excluded and so on. Since the two class intervals contain 115 cm. It is customary to include 115 cm in the class interval 115 cm  120 cm, which is the lower limit of the class interval.
Tally mar ks In the t he above example 7.1, 7.1, the height 110 110 cm, 112 112 cm belongs to in the t he class inter interval val 110  115 115.. We enter   tally marks. marks . Count the tally marks mark s and enter 2 as the frequency fr equency in the frequency f requency column. column.
If ve tally marks are to be made we mark four tally marks rst and the fth f th one one is marked across, so that     represe represents nts a cluster of of ve ve tally marks. To represent seven, we use a cluster cluster of ve tally marks and then add two more tally marks as shown
 .
221
Chapter 7 Frequency Table A table which represents the data in the form of three columns,
ﬁrst
column
showing the variable (Number) and the second column showing the values of the variable (Tally mark) and the third column showing their frequencies is called a frequency table (Refer table 7.3).
If the values of the variabl var iablee are given using different classes classes and the frequencies are marked against the respective classes, we get a frequency distribution. distribution . All the frequencies are added and the number is written as the total frequency for the entire intervals. This must match the total number of data given. The above process of forming a frequency f requency table table is called tabulation of data
.
Now we have the following table for the above data. (Example 7.1) Inclusive form Class Interval
Tally Marks
Frequency
110  114

2
115  119

3
120  124
 
6
125  129

5
130  134

4
Total
20
Table 7.1
Exclusivee form Exclusiv for m Class Interval
Tally Marks
Frequency
110  115

2
115  120

3
120  125
 
6
125  130

5
130  135

4
Total
20
Table 7.2
222
Data Handling Fr eque equency ncy table for for an u ngr ouped dat a Ex ample 7.2
Construct a frequency table for the following following data. 5, 1, 3, 4, 2, 1, 3, 5, 4, 2 1, 5, 1, 3, 2, 1, 5, 3, 3, 2. Solution:
From the the data, we observe the numbers numbers 1, 2, 3, 3, 4 and 5 are repeated. Hence
under the number column, write the ve numbers 1, 2, 3, 4, and 5 one below the other. Now read the number and put the tally mark in the tally mark column column against the number. In the same way put the tally mark till the last number. Add Add the tally marks against the numbers 1, 2, 3, 4 and 5 and write the total in the corresponding frequency column. Now, add all the numbers under the frequency column and write it against the total. Nu m b er
Ta lly M a r k s
F r eq u en cy
1 2 3 4
   
5 4 5 2
5
 Total
4 20
Table 7.3
In the formation of Frequency distribution for the given data values, we should (i)
select a suitable number of of classes, not very small and also not very large.
(ii)
take a suitable class  inter interval val (o (orr class width) width) and
(iii) present the classes with increasing values values without without any gaps between classes.
Fr eque equency ncy table for for a grouped dat a Ex ample 7.3
The following data relate to mathematics marks obtained by 30 students in standard VII. Prepare a frequency table for the data. 25, 67, 78, 43, 21, 17, 49, 54, 76, 92, 20, 45, 86, 37, 35 60, 71, 49, 75, 49, 32, 67, 15, 82, 95, 76, 41, 36, 71, 62 Solution:
The minimum marks obtained is 15. 15. The maximum marks ma rks obtained is 95. 223
Chapter 7 Range = Maximum value – Minimum value = 95 – 15 = 80 Choose 9 classes with a class interval of 10. as 10  20, 20  30, g ,90  100. The following is the frequency table. C la ss I n t er va l (M a r k s)
Ta lly M a r k s
F r eq u en cy
10  20 20  30
 
2 3
30  40 40  50 50  60 60  70
   
4 5 2 4
70  80 80  90 90  100
    Total
6 2 2 30
Table 7.4
7.2 Continuous grouped Frequency distribution Table To nd the class limits in continuous grouped frequency distribution.
Steps to do (i) Find the difference between betwee n the upper limit of the rst class and lower lower limit of the second class. (ii) Divide the differe difference nce by 2. Let the answer be x. (iii) Subtrac Subtractt ‘ x lower wer limits of all the t he class intervals. x’ from lo (iv) Add ‘ x x’ to all the upper limits of all the class intervals. Now the new limits will be true class limits. Ex ample 7.4
Form the frequency distribution table for the following data which gives the ages of persons who watched a particular channel on T.V. Class Interval (Age) Number of persons
10 19
20 29
30  39
40  49
50  59
60  69
45
60
87
52
25
12
224
Data Handling Solution:
In this table, the classes given here have gaps. Hence we rewrite the classes using the exclusive method.
Difference between upper limits of rst class and lower limits of second class = 20 – 19 19 = 1 Divide the difference by 2 then, x
=
1 = 0.5 2
Now subtra subtract ct 0.5 0.5 from lower lower limits and add 0.5 0.5 to the upper limits. Now we get continuous continuous frequency distribution distribution table with with true tr ue class class limits. Class Interval (Age)
Frequency (Number of persons)
9.5  19.5
45
19.5  29.5
60
29.5  39.5
87
39.5  49.5
52
49.5  59.5
25
59.5  69.5
12 Table 7.5 Exercise 7.1
1. Choose the correct answer. i) The difference between the highest and lowest value of the variable in the given data. is called. (A)) Frequency (A
(B) Class limit
(C) Class inter interval val
(D) Range
ii) The marks marks scored by by a set of students in a test are 65, 97, 97, 78, 49, 23, 48, 59, 98. The range for this data is (A)) 90 (A
(B) 74
(C) 73
(D) 75
iii) The range of the the rst rst 20 natural numbers is (A) 18
(B) 19
(C) 20
(D) 21
iv) The lower lower limit limit of the class class interval interval 20 20  30 is (A)) 30 (A
(B) 20
(C) 25
(D) 10
v) The upper upper of the class class interval interval 50  60 is (A)) 50 (A
(B) 60
(C) 10
225
(D) 55
Chapter 7 2. Construct a frequency table for each of the following data: 10, 15, 13, 12, 14, 11, 11, 12, 13, 15 11, 13, 12, 15, 13, 12, 14, 14, 15, 11 3. In the town there there were were 26 patients in a hospital. The number of tablets given to them is given below. below. Draw a frequency table for the data. 2, 4, 3, 1, 2, 2, 2, 4, 3, 5, 2, 1, 1, 2 4, 5, 1, 2, 5, 4, 3, 3, 2, 1, 5, 4. 4. The number number of savings savings book accounts accounts opened opened in a bank during 25 weeks are are given as below. Find a frequency distribution for the data: 15, 25, 22, 20, 18, 15, 23, 17, 19, 12, 21, 26, 30 19, 17, 14, 20, 21, 24, 21, 16, 22, 20, 17, 14 5. The weight (in kg) 20 persons are given below. below. 42, 45, 51, 55, 49, 62, 41, 52, 48, 64 52, 42, 49, 50, 47, 53, 59, 60, 46, 54 Form the frequency table by taking taking class intervals 40  45, 45  50, 50  55, 55  60 60 and 60  65. 6. The marks obtained by 30 students of a class in a mathematics test are given below. 45, 35, 60, 41, 8, 28, 31, 39, 55, 72, 22, 75, 57, 33, 51 76, 30, 49, 19, 13, 40, 88, 95, 62, 17, 67, 50, 66, 73, 70 Form the grouped frequency table: 7. Form a continuous frequency distributio distribution n table from the given data. Class Interval (weight in kg.)
21  23
24  26
27  29
30  32
33  35
36  38
Frequency (Number of children)
2
6
10
14
7
3
8. The following data gives the heights of trees in a grove. Form a continuous
frequency distribution dist ribution table. table. Class Interval (Height in metres)
2 4
57
8  10
11  13
14  16
Frequency (Number of trees)
29
41
36
27
12
226
Data Handling 7.3 Mean Median, Median , Mode of un ungr grouped ouped dat a Ar ithmetic mean We use the word ‘average’ in our day to day life. Poovini spends on an average of about 5 hours daily for her studies. In the month of May the average average temperature at Chennai is 40 degree celsius. What do the above statement tell us? Poovini usually studies for 5 hours. On some days, she may study for less number of hours and on the other day she may study longer. The average temperature of 40 degree celsius, means that, the temperature at the month of May in chennai is 40 degree celsius. Some times it may be less than 40 degree celsius and at other time it may be more than 40 degree celsius. Average lies between the highest and the lowest value of the given data. Rohit gets gets the fo following llowing marks in different dif ferent subjects subjects in an examination. 62, 84, 92, 98, 74
In order to get the average marks scored by him in the examination, we rst add up all the marks obtained by him in different subjects. 62 + 84 + 92 + 98 + 74 = 410. and then divide the sum by the total number of subjects. (i.e. 5) The average marks scored by Rohit =
410 = 82. 5
This number help helpss us to understand the general level of of his academic achievement achievement and is referred to as mean. `
The average or arithmetic mean or mean is dened as follows.
Mean =
Sum of all observation observations s Total number of obse observation rvations s
Ex ample 7.5
Gayathri studies for 4 hours, 5 hours and 3 hours respective respectively ly on 3 consecutive days. How many hours did she study daily on an average? Solution:
Average study time =
Total number of study hours Number of days for which she studied.
227
Chapter 7 = =
4+5+3 3 12 3
hours
= 4 hours per day day.. Thus we can say that Gayathri studies st udies for 4 hours daily on an average. Ex ample 7.6
The monthly income of 6 families are ` 3500, ` 2700, ` 3000, ` 2800, ` 3900 and ` 2100. Find the mean income. Solution:
Average monthly income = = =
Total income income of 6 familes familes Number of families
` 3500 + 2700 + 3000 + 2800 + 3900 + 2100 6 18000 ` 6
= ` 3,000. Ex ample 7.7 7.7
The mean price of 5 pens is ` 75 75.. What is the total cost of 5 pens? Solution:
Mean =
Total cost cost of 5 pens Number of pens
Total cost of 5 pens = Mean
#
Number of pens
= ` 75 # 5 = ` 37 375 5
Median Consider a group of 11 students with the following height (in cm) 106, 110, 123, 125, 115, 120, 112, 115, 110, 120, 115. The Physical EducationTeacher Mr. Gowtham wants to divide the students into two groups so that each group has equal number of students. One group has height lesser than a particular par ticular height height and the other group has student with height greater than the particular par ticular height. Now,, Mr. Gowtham arranged Now a rranged the students according to their height in ascending order. 106, 110, 110, 112, 115, 115,115, 120, 120, 123, 125 228
Data Handling The middle value in the data is 115 because this th is value divides the students into two equal groups of of 5 students each. This values is called as median. Median refers refers to the value 115 which lies in the middle of the data.Mr. Gowtham decides to keep the middle student as a referee in the game. Median is dened dene d as the middle value of the data when the data is arranged
in ascending or descending order. Find the median of the following: 40, 50, 30, 60, 80, 70 Arrange Arr ange the given data in ascending order. 30, 40, 50, 60, 70, 80.
Find the actual distance between your school and house. Find the median of the place.
Here the number number of terms is 6 which which is even. even. So the third and fourth fourth terms ter ms are middle terms. The average value of of the two terms ter ms is the median. (i.e)) (i.e
Median =
50 + 60 110 = = 55. 2 2
(i) When the number of observations is odd, the middle number is the median. (ii) Wh When en the nu mber of obse obserr vations is is even, even, the median media n is the avera ge of of the two middle numbers. Ex ample 7.8
Find the median of the following data. 3, 4, 5, 3, 6, 7, 2. Solution:
Arrange Arr ange the data in ascending a scending order. order.
line represents the median.
2, 3, 3, 4, 5, 6, 7 The number of observation is 7 which is odd. `
In highways, the yellow
The middle value 4 the median.
Ex ample 7.9 7.9
Find the median of the data 12, 14, 25, 23, 18, 17, 24, 20. Solution:
Arrange the data in ascending order 12, 14, 17, 18, 20, 23, 24, 25. 229
Chapter 7 The number of observation is 8 which is even. `
Median is the average of the two middle terms 18 and 20. Median =
18 + 2 0 38 = = 19 2 2
Ex ample 7.10 7.10
Find the median of the ﬁrst 5 prime numbers. Solution:
The ﬁrst ﬁve prime numbers are 2, 3, 5, 7, 11. The number of observation is 5 which is odd. `
The middle value 5 is the median.
Mode Look at the following example, Mr. Raghavan, the owner of a ready made dress shop says that the most popular size of shirts he sells is of size 40 cm. Observe that here also, the owner is concerned about the number of shirts of different sizes sold. He is looking at the shirt size that is sold, the most . The highest occurring occurr ing event event is the sale of size size 40 cm. This value is called the the mode of the data. Mode is the variable which occurs most frequently in the given data.
Mode of Large data Putting the same observation together and counting them is not easy if the number of observation observation is large. large. In such cases we tabulate tabulate the data. Ex ample 7.11 7.11
Following are the margin of victory in the foot ball matches of a league. 1, 3, 2, 5, 1, 4, 6, 2, 5, 2, 2, 2, 4, 1, 2, 3, 2, 3, 2, 1, 1, 1, 2, 3, 2, 6, 4, 3, 2, 1, 1, 1, 4, 2, 1, 5, 3, 4, 4, 2, 1, 2. Find the mode of this data. dat a. Solution: Margin of victory
Tally Marks
Number of Matches
1
 
9
2 3 4
     
14 7 5
5 6
  Total
3 2 40
Table 7.6 230
Data Handling Now we we quickly say that ‘2’ ‘2’ is the mode. Since 2 has occur red the more number of times, then the t he most of the matches have been won with a victory margin margi n of 2 goals. goals. Ex ample 7.12 7.12
Find the mode of the following data.
Find the mode of the transport in i n your place.
3, 4, 5, 3, 6, 7 Solution:
3 occurs the t he most number of times. `
Mode of the data is 3.
Ex ample 7.13 7.13
Find the mode of the following data. 2, 2, 2, 3, 3, 4, 5,5, 5, 6,6, 8 Solution:
2 and 5 occur 3 times. `
Mode of the data is 2 and 5.
Ex ample 7.14 7.14
Find the mode of the ower.
Find the mode of the following data 90, 40, 68, 94, 50, 60. Solution:
Here there are no frequently occurring values. Hence this data has no mode. mode. Ex ample 7.15 7.15
The number of children in 20 families are 1, 2, 2, 1, 2, 1, 3, 1, 1, 3 1, 3, 1, 1, 1, 2, 1, 1, 2, 1. Find the mode. Solution: Number of Children 1 2
   
Number of Families 12 5
 Total
3 20
Tally Marks
3
Table 7.7
12 families have 1 child only only,, so the mode of of the data is 1.
231
Chapter 7 Exercise: 7.2
1. Choose the correct answer: i) The arithmetic arithmetic mean of 1, 3, 5, 7 and 9 is (A)) 5 (A
(B) 7
(C) 3
(D) 9
ii) The average marks of 5 children is 40 then their total mark is (A)) 20 (A
(B) 200
(C) 8
(D) 4
(C) 30
(D) 10
(C) 7
(D) 14
(C) 7
(D) 2
iii) The median of 30,50, 40, 10, 20 is (A)) 40 (A
(B) 20
iv) The median of 2, 4, 6, 6, 8, 10, 12 12 is (A)) 6 (A
(B) 8
v) The mode of 3, 3, 4, 7, 4, 3, 2, 4 is (A)) 3 (A
(B) 4
2. The marks in mathemati mathematics cs of 10 students students are 56, 48, 58, 60, 54, 76, 84, 92, 82, 98. Find the range and arithmetic mean 3. The weights of 5 people are 72 kg, 48 kg, 51 kg, 69 kg, 67 kg. Find the mean of their weights. 4. Two vessels contain 30 litres and 50 litres of milk separately. What is the capacity of the vessels if both share the milk equally? 5. The maximum temperature in a city on 7 days of a certain week was 34.8°C, 38.5°C, 33.4°C, 34.7°C, 35.8°C, 32.8°C, 34.3°C. Find the mean temperature temperature for the week. 6. The mean weight of 10 boys boys in a cricket cricket team is 65.5 65.5 kg. What What is the total total weight of 10 boys. 7. Find the median of the following data. 6, 14, 5, 13, 11, 7, 8 8. The weight of 7 chocolate bars in grams are 131, 132, 125, 127, 130, 129, 133. Find the median. 9. The runs scored scored by a batsman batsman in 5 innings innings are 60, 100, 78, 54, 49. Find the median. 10. Find the median median of the rst seven natural natural numbers. numbers. 11. Pocket money received by 7 students is given below. below. `
42, ` 22, ` 40, ` 28, ` 23, ` 26, ` 43. Find the median.
12. Find the mode of the given data. 3, 4, 3, 5, 3, 6, 3, 8, 4.
232
Data Handling 13. Twelve eggs collected in a farm have the following weights. 32 gm,40 gm, 27 gm, 32 gm, 38 gm, 45 gm, 40 gm, 32 gm, 39 gm, 40 gm, 30 gm, 31 gm, Find the mode of the above data. 14. Find the mode of the following data. 4, 6, 8, 10, 12, 14 15. Find the mode of the following data. 12, 14, 12, 16, 15, 13, 14, 18, 19, 12, 14, 15, 16, 15, 16, 16, 15, 17, 13, 16, 16, 15, 13, 15, 17, 15, 14, 15, 13, 15, 14.
1. An Any y co colllec ecttion of information in the fo forrm of numer eriica cal l gures giving the requiredinformationiscalleddata. 2. Th The era raw wda data tais isa an nun unpr proc oces essed seda and ndu unc ncla lass ssi ied edd dat ata. a. 3. The Thed data atawh which ichis isa arr rran anged gedi ing ngro roup ups( s(or orcla classe ssess)i )isc scal alled leda agr grou ouped pedd data ata.. 4. The Then num umber berof oft tim imes esap apart articu icula larv rval alue uerep repea eats tsits itself elfis isca calllled edits itsfr frequ equency ency. . 5. Ra Rang nge e= =Hi Hig ghe hest stv val alue ue– –L Lo owes est tva valu lue. e. 6. The Thediff differen erence cebetwe between enthe theupp upper erand andthe thelow lower erlim limit itis iscal called ledthe thewid width thof ofthe the classinterval. 7. Ave vera rage gelilies esbetw between eenth theh ehig ighes hesta tand ndth thel elow owest estv val alue ueof ofth theg egiv iven enda data ta.. 8. Mean =
sum of all the observatio observations ns total number of observations
9. Med Media iani nisd sden ened edas asth them emid iddl dlev eval alue ueof ofth thed edata ata,w ,when henth thed edata atais isar arra rang nged ed inascendingordescendingorder. 10. Mode Modeis isthe thevar variab iablew lewhich hichoccu occursmo rsmostfr stfreque equentl ntlyi yinth nthegi egiven venda data. ta.
233
24 5
9 7
14 3
77 4
1 4
22 27 4 15
4 5
3 5
2 7
7 12
7 12
7 18
1 5
1 4
9 16
29 40 12
1 2
234
1 2
2 63 9 2 1 8
1 4
1 2
48 35 13 16
7 5
9 4
7 10
4 9
1 6
1 12
2 33
5 7 1 15
1 54
8 5
35 36
 20 15 48 28
 19 15 47 28
46 28
7 12
 18 15
 17 15
11 16
7 6
6 6
3 8
18 5
24 13
3 5
 43 21
5 3
1 2
 12 13
13 3
24 7
 13 30
9 44
5 16
 69 26
 41 60
1 27
1 12
2 35
1 4
19 12
3 2
7 11
1 2
7 11
7 210
41 42
4 5 3 4
4 6
45 28
3 4
7 4
5 6
17 20 9 10
235
19 42
23 20
 43 28 3 4
17 40
7 132
 72 25
 35 169
 15 4 16 81 9 7
7 24 26
3 2
 12 11
98 125
8 7
66
44 375 3 43
3 2
106
236
2 9 45 28
237
4 xy 3
1 2
1 2
238
25 100
25 200
33 100
70 100
82 100
1 3
3 10
1 200
9 100
3 4
1 400
239
1 40
2 3
4 5
240
241
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+
+
+
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+
+ +
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242
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+
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+
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+
+
+
+ +
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+
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+
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243
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244