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# Class 9 SELINA Solutions Maths Chapter 20 - Area and Perimeter of Plane Figures

## Area and Perimeter of Plane Figures Exercise Ex. 20(A)

### Solution 1(a)

Correct option: (iii) four times

If length and corresponding altitude is doubled, then

### Solution 1(b)

Correct option: (iv) one-fourth

If each side is halved, then

### Solution 1(c)

Correct option: (iii) 6 cm

### Solution 1(d)

Correct option: (iii) 72 cm2

The area of a right-angled triangle is equal to half the product of the sides containing the right angle.

Therefore,

### Solution 1(e)

Correct option: (i) 54 cm2

### Solution 2

Since the sides of the triangle are 18cm,24cm and 30cm respectively.

Hence area of the triangle is

Again

Hence

### Solution 3

Let the sides of the triangle are

a=3x

b=4x

c=5x

Given that the perimeter is 144 cm.

hence

The sides are a=36 cm, b=48 cm and c=60 cm

Area of the triangle is

### Solution 4

(i)

Area of the triangle is given by

(ii)

Again area of the triangle

### Solution 6

Since the perimeter of the isosceles triangle is 36cm and base is 16cm. hence the length of each of equal sides are

Here

It is given that

Let 'h' be the altitude of the isosceles triangle.

Since the altitude from the vertex bisects the base perpendicularly, we can apply Pythagoras Theorem.

Thus we have,

### Solution 7

It is given that

Let 'a' be the length of an equal side.

Hence perimeter=

From ,

Area of

Area of

Now

### Solution 9

Given , AB = 8 cm, AD = 10 cm, BD = 12 cm, DC = 13 cm and DBC = 90o

Hence perimeter=8+10+13+5=36cm

Area of

Area of

Now

### Solution 11

(i)

(ii) Consider the following figure.

(iii)

### Solution 12

Let the height of the triangle be x cm.

Equal sides are (x+4) cm.

According to Pythagoras theorem,

Hence perimeter=

Area of the isosceles triangle is given by

Here a=20cm

b=24cm

hence

### Solution 13

Each side of the triangle is

Hence the area of the equilateral triangle is given by

The height h of the triangle is given by

### Solution 14

The area of the triangle is given as 150sq.cm

Hence AB=15cm,AC=20cm and

### Solution 15

Let the two sides be x cm and (x-3) cm.

Now

Hence the sides are 12cm, 9cm and

The required perimeter is 12+9+15=36cm.

## Area and Perimeter of Plane Figures Exercise Ex. 20(B)

### Solution 1(a)

Correct option: (ii)

When two diagonals of a quadrilateral cut each other at right angles, then

### Solution 1(b)

Correct option: (i)

### Solution 1(c)

Correct option: (iii) 256 cm2

Therefore,

### Solution 1(d)

Correct option: (iii) 324 cm2

### Solution 1(e)

Correct option: (iv) 36 cm

### Solution 4

Consider the figure:

From the right triangle ABD we have

The area of right triangle ABD will be:

Again from the equilateral triangle BCD we have

Therefore the area of the triangle BCD will be:

Hence the area of the quadrilateral will be:

### Solution 5

The figure can be drawn as follows:

Here ABD is a right triangle. So the area will be:

Again

Now BCD is an isosceles triangle and BP is perpendicular to BD, therefore

From the right triangle DPC we have

So

Hence the area of the quadrilateral will be:

### Solution 6

Let the width be x and length 2x km.

Hence

Hence the width is 100m and length is 200m

The required area is given by

### Solution 7

Length of the laid with grass=85-5-5=75m

Width of the laid with grass=60-5-5=50m

Hence area of the laid with grass is given by

### Solution 8

Area of the rectangle is given by

Let h be the height of the triangle ,then

### Solution 9

Consider the following figure.

Thus the required area = area shaded in blue + area shaded in red

= Area ABPQ +  Area TUDC +  Area A'PUD' +  Area QB'C'T

= 2Area ABPQ + 2Area QB'C'T

=2(Area ABPQ +Area QB'C'T)

### Solution 10

Perimeter of the garden

Again, length of the garden is given to be 120 m. hence breadth of the garden b is given by

Hence area of the field

### Solution 11

Length of the rectangle=x

Width of the rectangle=

Hence its perimeter is given by

Again it is given that the perimeter is 4400cm.

Hence

Length of the rectangle=1400 cm = 14 m

### Solution 12

(i)

Length of the verandah=x+3

According to the question

(ii)

From the above equation

Length =3+3=6m

### Solution 13

Consider the following figure.

Thus, the area of the shaded portion

=Area(ABCD) + Area(EFGH) - Area(IJKL)…(1)

### Solution 14

First we have to calculate the area of the hall.

We need to find the cost of carpeting of 80 cm = 0.8 m wide carpet, if the rate of carpeting is Rs. 25. Per metre.

Then

### Solution 15

Let a be the length of each side of the square.

Hence

Hence

And

### Solution 16

Consider the following figure.

(i)

The length of the lawn = 30 - 2 - 2 = 26 m

The breadth of the lawn = 12 - 2 = 10 m

(ii)

The orange shaded area in the figure is the required area.

Area of the flower bed is calculated as follows:

### Solution 17

Number of tiles required

Fraction of floor uncovered=

### Solution 18

Since

Hence the distance between the shorter sides is 16m.

### Solution 19

At first we have to calculate the area of the triangle having sides, 28cm, 26cm and 30cm.

Let the area be S.

By Heron's Formula,

Area of a Parallelogram = 2 × Area of a triangle

= 2 × 336

= 672 cm2

We know that,

Area of a parallelogram = Height × Base

672 = Height × 26

Height = 25.84 cm

the distance between its shorter sides is 25.84 cm.

### Solution 20

(i)

We know that

Here A=216sq.cm

AC=24cm

BD=?

Now

(ii)

Let a be the length of each side of the rhombus.

(iii)

Perimeter of the rhombus=4a=60cm.

### Solution 21

Let a be the length of each side of the rhombus.

(i)

It is given that,

AC=24cm

We have to find BD.

Now

Hence the other diagonal is 10cm.

(ii)

### Solution 22

Let a be the length of each side of the rhombus.

We know that,

### Solution 23

The diagram is redrawn as follows:

Here

AF=1.2m,EF=0.3m,DC=0.6m,BK=1.8-0.6-0.3=0.9m

Hence

### Solution 24

Here we found two geometrical figure, one is a triangle and other is the trapezium.

Now

hence area of the whole figure=150+240=630sq.m

### Solution 25

We can divide the field into three triangles and one trapezium.

Let A,B,C be the three triangular region and D be the trapezoidal region.

Now

Area of the figure=Area of A+ Area of B+ Area of C+ Area of D

=3900+1250+312.5+2062.5

=7525sq.m

## Mean and Median (For Ungrouped Data Only) Exercise Ex. 20(C)

### Solution 1(a)

Correct option: (i) 77 cm2

In right-angled ΔABC,

Diameter of semi-circle BPC = BC = 14 cm

⇒ Radius of semi-circle BPC, r = 7 cm

### Solution 1(b)

Correct option: (ii) doubled

Diameter of a circle = 14 cm

If diameter is doubled, then

### Solution 1(c)

Correct option: (iii) 42 cm2

Radius of a quadrant OCA = Side of a square = 14 cm

Then,

Now, area of shaded portion = Area of square OCBA - Area of quadrant OCA

= (196 - 154) cm2

= 42 cm2

### Solution 1(d)

Correct option: (iii) four times

Let the radius of a circle = r

Then,

### Solution 1(e)

Correct option: (i) 1848 cm2

Therefore,

area of shaded portion = Area of circle with radius 28 cm - Area of circle with radius 14 cm

= (2464 - 616) cm2

= 1848 cm2

### Solution 2

Let r be the radius of the circle.

(i)

= 88 cm

(ii)

= 616 cm2

### Solution 4

Let r be the radius of the circle.

### Solution 5

Circumference of the first circle

Circumference of the second circle

Let r be the radius of the resulting circle.

### Solution 6

Circumference of the first circle

Circumference of the second circle

Let r be the radius of the resulting circle.

Hencearea of the circle

### Solution 7

Let the area of the resulting circle be r.

Hence the radius of the resulting circle is 20cm.

### Solution 8

Area of the circle having radius 85m is

Let r be the radius of the circle whose area is 49times of the given circle.

Hence circumference of the circle

### Solution 9

Area of the rectangle is given by

For the largest circle, the radius of the circle will be half of the sorter side of the rectangle.

Hence r=21cm.

Hence

### Solution 10

Area of the square is given by

Since there are four identical circlesinside the square.

Hence radius of each circle is one fourth of the side of the square.

Area remaining in the cardboard is

### Solution 11

Let the radius of the two circles be 3r and 8r respectively.

According to the question

Hence radius of the smaller circle is

Area of the smaller circle is given by

### Solution 12

Let the diameter of the three circles be 3d, 5d and 6d respectively.

Now

### Solution 14

Let r be the radius of the circular park.

Radius of the outer circular region including the path is given by

Area of that circular region is

Hence area of the path is given by

### Solution 19

Time interval is

Area covered in one 60 minutes=

Hence area swept in 35 minutes is given by

### Solution 20

Let R and r be the radius of the big and small circles respectively.

## Area and Perimeter of Plane Figures Exercise Test Yourself

Since AB=AC and

Now

### Solution 2

Let x be the width of the footpath.

Then

Again it is given that area of the footpath is 360sq.m.

Hence

Hence width of the footpath is 3m.

### Solution 3

Area of the square is 484.

Let a be the length of each side of the square.

Now

Hence length of the wire is=4x22=88m.

(i)

Now this 88m wire is bent in the form of an equilateral triangle.

(ii)

Let x be the breadth of the rectangle.

Now

Hence area=16x28=448m2

### Solution 4

(i)

Again

(ii)

(iii)

For the triangle EBC,

S=19cm

Let h be the height.

(iv)

In the given figure, we can observe that the non-parallel sides are equal and hence it is an isosceles trapezium.

Therefore, let us draw DE and CF perpendiculars to AB.

Thus, the area of the parallelogram is given by

### Solution 5

Let b be the breadth of rectangle. then its perimeter

Again

Hence its length is 20cm and width is 15cm.

### Solution 6

Let b be the width of the rectangle.

Again perimeter of the rectangle is 104m.

Hence

Hence

length=32m

width=20m.

### Solution 7

Let a be the length of the sides of the square.

According to the question

Hence sides of the square are 12cm each and

Length of the rectangle =2a=24cm

Width of the rectangle=a+6=18cm.

### Solution 9

Length of the wall=45+2=47m

Breath of the wall=30+2=32m

Hence area of the inner surface of the wall is given by

### Solution 10

Let a be the length of each side.

Hence length of the wire=96cm

(i)

For the equilateral triangle,

(ii)

Let the adjacent side of the rectangle be x and y cm.

Since the perimeter is 96 cm, we have,

Hence

Hence area of the rectangle is = 26 x 22 = 572 sq.cm

### Solution 11

Let 'y' and 'h' be the area and the height of the first parallelogram respectively.

Let 'height' be the height of the second parallelogram

base of the first parallelogram=cm

base of second parallelogram=cm

### Solution 14

Let the radius of the field is r meter.

Therefore circumference of the field will be: meter.

Now the cost of fencing the circular field is 52,800 at rate 240 per meter.

Therefore

Thus the radius of the field is 35 meter.

Now the area of the field will be:

Thus the cost of ploughing the field will be:

### Solution 15

Let r and R be the radius of the two circles.

Putting the value of r in (2)

Hence the radius of the two circles is 3cm and 7cm respectively.

### Solution 17

Since the diameter of the circle is the diagonal of the square inscribed in the circle.

Let a be the length of the sides of the square.

Hence

Hence the area of the square is 98sq.cm.

### Solution 19

Let the sides of the triangle be

a = 12x

b = 5x

c = 13x

Given that the perimeter = 450 cm

12x + 5x + 13x = 450

30x = 450

x = 15

Hence, the sides of a triangle are

a = 12x = 12(15) = 180 cm

b = 5x = 5(15) = 75 cm

c = 13x = 13(15) = 195 cm

Now,

### Solution 20

Let the sides of the triangle be

a = 26 cm, b = 28 cm and c = 30 cm

Now,

### Solution 21

Construction: Draw CM AB

In right-angled triangle CMB,

BM2 = BC2 - CM2 = (15)2 - (9)2 = 225 - 81 = 144

BM = 12 m

Now, AB = AM + BM = 23 + 12 = 35 m

### Solution 22

Let the sides of two squares be a and b respectively.

Then, area of one square, S1 = a2

And, area of second square, S2 = b2

Given, S1 + S2 = 400 cm2

a2 + b2 = 400 cm2 ..(1)

Also, difference in perimeter = 16 cm

4a - 4b = 16 cm

a - b = 4

a = (4 + b)

Substituting the value of 'a' in (1), we get

(4 + b)2 + b2 = 400

16 + 8b + b2 + b2 = 400

2b2 + 8b - 384 = 0

b2 + 4b - 192 = 0

b2 + 16b - 12b - 192 = 0

b(b + 16) - 12(b + 16) = 0

(b +16)(b - 12) = 0

b + 16 = 0 or b - 12 = 0

b = -16 or b = 12

Since, the side of a square cannot be negative, we reject -16.

Thus, b = 12

a = 4 + b = 4 + 12 = 16

Hence, the sides of a square are 16 cm and 12 cm respectively.

### Solution 23

Diagonal of a square = 24 cm

### Solution 26

Let the radii of two circles be r1 and r2 respectively.

Sum of the areas of two circles = 130π sq. cm

πr12 + πr22 = 130π

r12 + r22 = 130 ….(i)

Also, distance between two radii = 14 cm

r1 + r2 = 14

r1 = (14 - r2)

Substituting the value of r1 in (i), we get

(14 - r2)2 + r22 = 130

196 - 28r2 + r22 + r22 = 130

2r22 - 28r2 + 66 = 0

r22 - 14r2 + 33 = 0

r22 - 11r2 - 3r2 + 33 = 0

r2 (r2 - 11) - 3 (r2 - 11) = 0

(r2 - 11) (r2 - 3) = 0

r2 = 11 or r2 = 3

r1 = 14 - 11 = 3 or r1 = 14 - 3 = 11

Thus, the radii of two circles are 11 cm and 3 cm respectively.