Rs Aggarwal 2018 for Class 10 Math Chapter 17 - Perimeter And Area Of Plane Figures

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Rs Aggarwal 2018 Solutions for Class 10 Math Chapter 17 Perimeter And Area Of Plane Figures are provided here with simple step-by-step explanations. These solutions for Perimeter And Area Of Plane Figures are extremely popular among Class 10 students for Math Perimeter And Area Of Plane Figures Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal 2018 Book of Class 10 Math Chapter 17 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggarwal 2018 Solutions. All Rs Aggarwal 2018 Solutions for class Class 10 Math are prepared by experts and are 100% accurate.

Page No 772:

Question 1:

Find the area of the triangle whose base measures 24 cm and the corresponding height measures 14.5 cm.

Answer:

Given: $base = 24 cm, correponding height = 14.5 cm$
Area of a triangle = $\frac{1}{2} \times base \times coresponding height$

$= \frac{1}{2} \times 24 \times 14.5 = 174 {cm}^{2}$

Page No 772:

Question 2:

Find the area of the triangle whose sides are 42 cm, 34 cm and 20 cm in length. Find the height corresponding to the longest side.

Answer:

Let the sides of the triangle be a = 20 cm, b = 34 cm and c = 42 cm.
Let s be the semi-perimeter of the triangle.
$s = \frac{1}{2} (a + b + c) s = \frac{1}{2} (20 + 34 + 42) s = 48 cm$

$Area of t h e triangle = \sqrt{s (s - a) (s - b) (s - c)} \Rightarrow \sqrt{48 (48 - 20) (48 - 34) (48 - 42)} \Rightarrow \sqrt{48 \times 28 \times 14 \times 6} \Rightarrow \sqrt{112896} \Rightarrow 336 {cm}^{2}$

Length of the longest side is 42 cm.

Area of a triangle = $\frac{1}{2} \times b \times h$
$\Rightarrow 336 = \frac{1}{2} \times 42 \times h \Rightarrow 672 = 42 h \Rightarrow \frac{672}{42} = h \Rightarrow h = 16 cm$

The height corresponding to the longest side is 16 cm.

Page No 772:

Question 3:

Find the area of the triangle whose sides are 18 cm, 24 cm and 30 cm. Also, find the height corresponding to the smallest side.

Answer:

Let the sides of triangle be a = 18 cm, b = 24 cm and c = 30 cm.
Let s be the semi-perimeter of the triangle.
$s = \frac{1}{2} (a + b + c) s = \frac{1}{2} (18 + 24 + 30) s = 36 cm$

$Area of a t riangle = \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{36 (36 - 18) (36 - 24) (36 - 30)} = \sqrt{36 \times 18 \times 12 \times 6} = \sqrt{46656} = 216 {cm}^{2}$
The smallest side is 18 cm long. This is the base.

Now, area of a triangle = $\frac{1}{2} \times b \times h$
$\Rightarrow 216 = \frac{1}{2} \times 18 \times h \Rightarrow 216 = 9 h \Rightarrow \frac{216}{9} = h \Rightarrow h = 24 cm$

The height corresponding to the smallest side is 24 cm.

Page No 772:

Question 4:

The sides of a triangle are in the ratio 5 : 12 : 13, and its perimeter is 150 m. Find the area of the triangle.

Answer:

Let the sides of a triangle be 5x m ,12x m and 13x m.

Since, perimeter is the sum of all the sides,
$5 x + 12 x + 13 x = 150 \Rightarrow 30 x = 150 o r, x = \frac{150}{30} = 5$

The lengths of the sides are:
$a = 5 \times 5 = 25 m b = 12 \times 5 = 60 m c = 13 \times 5 = 65 m Semiperimeter (s) of the triangle = \frac{Perimeter}{2} = \frac{25 + 60 + 65}{2} = \frac{150}{2} = 75 m Area of triangle = \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{75 (75 - 25) (75 - 60) (75 - 65)} = \sqrt{75 \times 50 \times 15 \times 10} = \sqrt{562500} = 750 m^{2}$

Page No 772:

Question 5:

The perimeter of a triangular field is 540 m, and its sides are in the ratio 25:17:12. Find the area of the field. Also, find the cost of ploughing the field at ₹40 per 100 m².

Answer:

$Let the sides of the triangular field be 25 x, 17 x and 12 x . As, perimeter = 540 m \Rightarrow 25 x + 17 x + 12 x = 540 \Rightarrow 54 x = 540 \Rightarrow x = \frac{540}{54} \Rightarrow x = 10 So, the sides are 250 m, 170 m and 120 m . Now, semi - perimeter, s = \frac{250 + 170 + 120}{2} = \frac{540}{2} = 270 m So, area of the field = \sqrt{270 (270 - 250) (270 - 170) (270 - 120)} = \sqrt{270 \times 20 \times 100 \times 150} = \sqrt{3^{3} \times 10 \times 2 \times 10 \times 10^{2} \times 3 \times 5 \times 10} = 3^{2} \times 10^{3} = 9000 m^{2} Also, the cost of ploughing the field = \frac{9000 \times 40}{100} = ₹ 3, 600$

Page No 772:

Question 6:

The perimeter of a right triangle is 40 cm and its hypotenuse measures 17 cm. Find the area of the triangle.

Answer:

The perimeter of a right-angled triangle = 40 cm
Therefore , a+b+c= 40 cm
Hypotenuse = 17 cm
Therefore, c = 17 cm
a+b+c= 40 cm
$\Rightarrow$ a+b+17 = 40
$\Rightarrow$ a+b = 23
$\Rightarrow$ b = 23 $-$ a...........(i)
Now, using Pythagoras' theorem, we have:
$a^{2} + b^{2} = c^{2} \Rightarrow a^{2} + (23 - a)^{2} = 17^{2} \Rightarrow a^{2} + 529 - 46 a + a^{2} = 289 \Rightarrow 2 a^{2} - 46 a + 529 - 289 = 0 \Rightarrow 2 a^{2} - 46 a + 240 = 0 \Rightarrow a^{2} - 23 a + 120 = 0 \Rightarrow (a - 15) (a - 8) = 0 \Rightarrow a = 15 o r a = 8$

Substituting the value of a=15, in equation(i) we get:
b = 23-a
= 23 - 15
= 8 cm

If we had chosen $a = 8 cm$ , then, $b = 23 - 8 = 15 cm$

In any case,
$Area of a t riangle = \frac{1}{2} \times base \times height = \frac{1}{2} \times 8 \times 15 = 60 {cm}^{2}$

Page No 772:

Question 7:

The difference between the sides at right angle in a right-angled triangle is 7 cm. The area of the triangle is 60 cm². Find its perimeter.

Answer:

Given:
Area of the triangle = $60 {cm}^{2}$
Let the sides of the triangle be a, b and c, where a is the height, b is the base and c is hypotenuse of the triangle.
$a - b = 7 cm$
a = 7 + b.......(1)
$Area of triangle = \frac{1}{2} \times b \times h$
$\Rightarrow 60 = \frac{1}{2} \times b \times (7 + b) \Rightarrow 120 = 7 b + b^{2} \Rightarrow b^{2} + 7 b - 120 = 0 \Rightarrow (b + 15) (b - 8) = 0 \Rightarrow b = - 15 or 8$

Side of a triangle cannot be negative.
Therefore, b = 8 cm.

Substituting the value of b = 8 cm, in equation (1):
a = 7+8 = 15 cm

Now, a = 15 cm, b = 8 cm

Now, in the given right triangle, we have to find third side.
$(Hyp)^{2} = (F i r s t side)^{2} + (S e c o n d side)^{2} \Rightarrow {Hyp}^{2} = 8^{2} + 15^{2} \Rightarrow {Hyp}^{2} = 64 + 225 \Rightarrow {Hyp}^{2} = 289 \Rightarrow Hyp = 17 cm$
So, the third side is 17 cm.

Perimeter of a triangle = $a + b + c$ .
Therefore, required perimeter of the triangle $= 15 + 8 + 17 = 40 cm$ .

Page No 772:

Question 8:

The lengths of the two sides of a right triangle containing the right angle differ by 2 cm. If the area of the triangle is 24 cm², find the perimeter of the triangle.

Answer:

Given:
Area of triangle = $24 {cm}^{2}$
Let the sides be a and b, where a is the height and b is the base of triangle.

$a - b = 2 cm$
a = 2 + b.......(1)

$Area of triangle = \frac{1}{2} \times b \times h$

$\Rightarrow 24 = \frac{1}{2} \times b \times (2 + b) \Rightarrow 48 = b + \frac{1}{2} b^{2} \Rightarrow 48 = 2 b + b^{2} \Rightarrow b^{2} + 2 b - 48 = 0 \Rightarrow (b + 8) (b - 6) = 0 \Rightarrow b = - 8 or 6$
Side of a triangle cannot be negative.

Therefore, b = 6 cm.

Substituting the value of b = 6 cm in equation(1), we get:
a = 2+6 = 8 cm

Now, a = 8 cm, b = 6 cm

In the given right triangle we have to find third side. Using the relation

$(Hyp)^{2} = (Oneside)^{2} + (Otherside)^{2} \Rightarrow {Hyp}^{2} = 8^{2} + 6^{2} \Rightarrow {Hyp}^{2} = 64 + 36 \Rightarrow {Hyp}^{2} = 100 \Rightarrow Hyp = 10 cm$

So, the third side is 10 cm.

So, perimeter of the triangle = a + b + c
= 8+6+10
=24 cm

Page No 772:

Question 9:

Each side of an equilateral triangle is 10 cm. Find

(i) the area of the triangle and (ii) the height of the triangle.

Answer:

$(i) The area of the equilateral triangle = \frac{\sqrt{3}}{4} \times {side}^{2} = \frac{\sqrt{3}}{4} \times 10^{2} = \frac{\sqrt{3}}{4} \times 100 = 25 \sqrt{3} {cm}^{2} or 25 \times 1.732 = 43.3 {cm}^{2} So, the area of the triangle is 25 \sqrt{3} {cm}^{2} or 43.3 {cm}^{2} . (ii) As, area of the equilateral triangle = 25 \sqrt{3} {cm}^{2} \Rightarrow \frac{1}{2} \times Base \times Height = 25 \sqrt{3} \Rightarrow \frac{1}{2} \times 10 \times Height = 25 \sqrt{3} \Rightarrow 5 \times Height = 25 \sqrt{3} \Rightarrow Height = \frac{25 \sqrt{3}}{5} = 5 \sqrt{3} or height = 5 \times 1.732 = 8.66 m ∴ The height of the triangle is 5 \sqrt{3} cm or 8.66 cm .$

Page No 772:

Question 10:

The height of an equilateral triangle is 6 cm. Find its area. [Take $\sqrt{3}$ = 1.73]

Answer:

Let the side of the equilateral triangle be x cm.

$As, the area of an equilateral triangle = \frac{\sqrt{3}}{4} {(side)}^{2} = \frac{x^{2} \sqrt{3}}{4} Also, the area of the triangle = \frac{1}{2} \times Base \times Height = \frac{1}{2} \times x \times 6 = 3 x So, \frac{x^{2} \sqrt{3}}{4} = 3 x \Rightarrow \frac{x \sqrt{3}}{4} = 3 \Rightarrow x = \frac{12}{\sqrt{3}} \Rightarrow x = \frac{12}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \Rightarrow x = \frac{12 \sqrt{3}}{3} \Rightarrow x = 4 \sqrt{3} cm Now, area of the equilateral triangle = 3 x = 3 \times 4 \sqrt{3} = 12 \sqrt{3} = 12 \times 1.73 = 20.76 {cm}^{2}$

Page No 772:

Question 11:

If the area of an equilateral triangle is $36 \sqrt{3} {cm}^{2}$ , find its perimeter.

Answer:

Area of equilateral triangle = $36 \sqrt{3} {cm}^{2}$

Area of equilateral triangle = $(\frac{\sqrt{3}}{4} \times a^{2})$ , where a is the length of the side.
$\Rightarrow 36 \sqrt{3} = \frac{\sqrt{3}}{4} \times a^{2} \Rightarrow 144 = a^{2} \Rightarrow a = 12 cm$

Perimeter of a triangle = 3a
$= 3 \times 12 = 36 cm$

Page No 772:

Question 12:

If the area of an equilateral triangle is $81 \sqrt{3} {cm}^{2}$ , find its height.

Answer:

Area of the equilateral triangle = $81 \sqrt{3} {cm}^{2}$
Area of an equilateral triangle = $(\frac{\sqrt{3}}{4} \times a^{2})$ , where a is the length of the side.
$\Rightarrow 81 \sqrt{3} = \frac{\sqrt{3}}{4} \times a^{2} \Rightarrow 324 = a^{2} \Rightarrow a = 18 cm$
Height of triangle = $\frac{\sqrt{3}}{2} \times a$

$= \frac{\sqrt{3}}{2} \times 18 = 9 \sqrt{3} cm$

Page No 772:

Question 13:

The base of a right-angled triangle measures 48 cm and its hypotenuse measures 50 cm. Find the area of the triangle.

Answer:

Base = 48 cm
Hypotenuse =50 cm
First we will find the height of the triangle; let the height be 'p'.
$\Rightarrow {(Hypotenuse)}^{2} = {(b a s e)}^{2} + p^{2} \Rightarrow 50^{2} = 48^{2} + p^{2} \Rightarrow p^{2} = 50^{2} - 48^{2} \Rightarrow p^{2} = (50 - 48) (50 + 48) \Rightarrow p^{2} = 2 \times 98 \Rightarrow p^{2} = 196 \Rightarrow p = 14 cm$

Area of the triangle= $\frac{1}{2} \times base \times height$

$= \frac{1}{2} \times 48 \times 14 = 336 {cm}^{2}$

Page No 772:

Question 14:

The hypotenuse of a right-angled triangle measures 65 cm and its base is 60 cm. Find the length of perpendicular and the area of the triangle.

Answer:

Hypotenuse = 65 cm
Base = 60 cm
In a right-angled triangle,
${(Hypotenuse)}^{2} = {(B ase)}^{2} + {(P erpendicular)}^{2} \Rightarrow {(65)}^{2} = {(60)}^{2} + {(perpendicular)}^{2} \Rightarrow {(65)}^{2} - {(60)}^{2} = {(perpendicular)}^{2} \Rightarrow {(perpendicular)}^{2} = (65 - 60) (65 + 60) \Rightarrow {(perpendicular)}^{2} = 5 \times 125 \Rightarrow {(perpendicular)}^{2} = 625 \Rightarrow perpendicular = 25 cm$

Area of triangle = $\frac{1}{2} \times base \times perpendicular$

$= \frac{1}{2} \times 60 \times 25 = 750 {cm}^{2}$

Page No 773:

Question 15:

Find the area of a right-angled triangle, the radius of whose circumcircle measure 8 cm and the altitude drawn to the hypotenuse measures 6 cm.

Answer:

Given: Radius = 8 cm
Height = 6 cm
Area=?
In a right-angled triangle, the centre of the circumcircle is the mid-point of the hypotenuse.

$Hypotenuse = 2 \times (radius of circumcircle) for a right triangle = 2 \times 8 = 16 cm So, h ypotenuse = 16 cm$
Now, base = 16 cm and height = 6 cm

Area of the triangle = $\frac{1}{2} \times b a s e \times h e i g h t$

$= \frac{1}{2} \times 16 \times 6 = 48 {cm}^{2}$

Page No 773:

Question 16:

Find the length of the hypotenuse of an isosceles right-angled triangle whose area is 200 cm². Also, find its perimeter. [Given: $\sqrt{2}$ = 1.41]

Answer:

In a right isosceles triangle, $base = height = a$

Therefore,
$Area of the triangle = \frac{1}{2} \times base \times height = \frac{1}{2} \times a \times a = \frac{1}{2} a^{2}$

Further, given that area of isosceles right triangle = 200 cm²

$\Rightarrow \frac{1}{2} a^{2} = 200 \Rightarrow a^{2} = 400 o r, a = \sqrt{400} = 20 cm$

In an isosceles right triangle, two sides are equal ('a') and the third side is the hypotenuse, i.e. 'c'

Therefore, c = $\sqrt{a^{2} + a^{2}}$

    $= \sqrt{2 a^{2}} = a \sqrt{2} = 20 \times 1.41 = 28.2 cm$

Perimeter of the triangle = $a + a + c$
                           $= 20 + 20 + 28.2$

                            = 68.2 cm

The length of the hypotenuse is 28.2 cm and the perimeter of the triangle is 68.2 cm.

Page No 773:

Question 17:

The base of an isosceles triangle measures 80 cm and its area is 360 cm². Find the perimeter of the triangle.

Answer:

Given :
Base = 80 cm
Area = 360 ${cm}^{2}$
Area of an isosceles triangle = $(\frac{1}{4} b \sqrt{4 a^{2} - b^{2}})$

$\Rightarrow 360 = \frac{1}{4} \times 80 \sqrt{4 a^{2} - 80^{2}} \Rightarrow 360 = 20 \sqrt{4 a^{2} - 6400} \Rightarrow 18 = 2 \sqrt{a^{2} - 1600} \Rightarrow 9 = \sqrt{a^{2} - 1600}$
Squaring both the sides, we get:
$\Rightarrow 81 = a^{2} - 1600 \Rightarrow a^{2} = 1681 \Rightarrow a = 41 cm$
Perimeter $= (2 a + b)$
$= [2 (41) + 80] = 82 + 80 = 162 cm$
So, the perimeter of the triangle is 162 cm.

Page No 773:

Question 18:

Each of the equal sides of an isosceles triangle measures 2 cm more than its height, and the base of the triangle measures 12 cm. Find the area of the triangle.

Answer:

Let the height of the triangle be h cm.
Each of the equal sides measures $a = (h + 2) cm$ and b = 12 cm (base).

Now,
Area of the triangle = Area of the isosceles triangle
$\frac{1}{2} \times base \times height = \frac{1}{4} \times b \sqrt{4 a^{2} - b^{2}}$

$\Rightarrow \frac{1}{2} \times 12 \times h = \frac{1}{4} \times 12 \times \sqrt{4 (h + 2)^{2} - 144} \Rightarrow 6 h = 3 \sqrt{4 h^{2} + 16 h + 16 - 144} \Rightarrow 2 h = \sqrt{4 h^{2} + 16 h + 16 - 144} On squaring both the sides, we get : \Rightarrow 4 h^{2} = 4 h^{2} + 16 h + 16 - 144 \Rightarrow 16 h - 128 = 0 \Rightarrow h = 8$

Area of the triangle = $\frac{1}{2} \times b \times h$
$= \frac{1}{2} \times 12 \times 8 = 48 {cm}^{2}$

Page No 773:

Question 19:

Find the area and perimeter of an isosceles right-angled triangle, each of whose equal sides measures 10 cm. [Given: $\sqrt{2}$ = 1.41]

Answer:

Let:
Length of each of the equal sides of the isosceles right-angled triangle = a = 10 cm
And,
Base = Height = a
$Area of isosceles right - angled triangle = \frac{1}{2} \times Base \times Height = \frac{1}{2} \times 10 \times 10 = 50 {cm}^{2}$

The hypotenuse of an isosceles right-angled triangle can be obtained using Pythagoras' theorem.

If h denotes the hypotenuse, we have:

$h^{2} = a^{2} + a^{2} \Rightarrow h = 2 a^{2} \Rightarrow h = \sqrt{2} a \Rightarrow h = 10 \sqrt{2} cm$

∴ Perimeter of the isosceles right-angled triangle = $2 a + \sqrt{2} a$
$= 2 \times 10 + 1.41 \times 10 = 20 + 14.1 = 34.1 cm$

Page No 773:

Question 20:

In the given figure, ∆ABC is an equilateral triangle the length of whose side is equal to 10 cm, and ∆DBC is right-angled at D and BD = 8 cm. Find the area of the shaded region. Take $\sqrt{3}$ = 1.732.

Answer:

Given:
Side of equilateral triangle ABC = 10 cm
BD = 8 cm
Area of $equilateral ∆ A B C = \frac{\sqrt{3}}{4} a^{2}$ (where a = 10 cm)
$Area of equilateral △ A B C = \frac{\sqrt{3}}{4} \times 10^{2} = 25 \sqrt{3} = 25 \times 1.732 = 43.30 {cm}^{2}$

$In the right ∆ B D C, we have : B C^{2} = B D^{2} + C D^{2} \Rightarrow 10^{2} = 8^{2} + C D^{2} \Rightarrow C D^{2} = 10^{2} - 8^{2} \Rightarrow C D^{2} = 36 \Rightarrow C D = 6$

Area of triangle $△$ BCD = $\frac{1}{2} \times b \times h$

                $= \frac{1}{2} \times 8 \times 6 = 24 {cm}^{2}$

Area of the shaded region = $Area of ∆ A B C - Area of ∆ B D C$
   = 43.30 $-$ 24
           = 19.3 cm²

Page No 780:

Question 1:

The perimeter of a rectangular plot of land is 80 m and its breadth is 16 m. Find the length and area of the plot.

Answer:

$As, perimeter = 80 m \Rightarrow 2 (length + breadth) = 80 \Rightarrow 2 (length + 16) = 80 \Rightarrow 2 \times length + 32 = 80 \Rightarrow 2 \times length = 80 - 32 \Rightarrow length = \frac{48}{2} ∴ length = 24 m Now, the area of the plot = length \times breadth = 24 \times 16 = 384 m^{2}$

So, the length of the plot is 24 m and its area is 384 m².

Page No 780:

Question 2:

The length of a rectangular park is twice its breadth, and its perimeter measures 840 m. Find the area of the park.

Answer:

Let the breadth of the rectangular park be $b$ .
∴ Length of the rectangular park $= l = 2 b$
Perimeter = 840 m

$\Rightarrow 840 = 2 (l + b) \Rightarrow 840 = 2 (2 b + b) \Rightarrow 840 = 2 (3 b) \Rightarrow 840 = 6 b \Rightarrow b = 140 m Thus, we have : l = 2 b = 2 \times 140 = 280 m$

$Area = l \times b = 280 \times 140 = 39200 m^{2}$

Page No 781:

Question 3:

One side of a rectangle is 12 cm long and its diagonal measures 37 cm. Find the other side and the area of the rectangle.

Answer:

One side of the rectangle = 12 cm
Diagonal of the rectangle = 37 cm

The diagonal of a rectangle forms the hypotenuse of a right-angled triangle. The other two sides of the triangle are the length and the breadth of the rectangle.

Now, using Pythagoras' theorem, we have:

${(one side)}^{2} + {(other side)}^{2} = {(hypotenuse)}^{2}$
$\Rightarrow {(12)}^{2} + {(other side)}^{2} = {(37)}^{2} \Rightarrow 144 + {(other side)}^{2} = 1369 \Rightarrow {(other side)}^{2} = 1369 - 144 \Rightarrow {(other side)}^{2} = 1225 \Rightarrow other side = \sqrt{1225} \Rightarrow other side = 35 cm$

Thus, we have:
Length = 35 cm
Breadth = 12 cm
$Area of the rectangle = 35 \times 12 = 420 {cm}^{2}$

Page No 781:

Question 4:

The area of a rectangular plot is 462 m² and its length is 28 m. Find the perimeter of the plot.

Answer:

Area of the rectangular plot = 462 m²
Length (l) = 28 m
$Area of a rectangle = Length (l) x Breadth (b) \Rightarrow 462 = 28 x b \Rightarrow b = 16.5 m$

Perimeter of the plot = $2 (l + b)$
$= 2 (28 + 16.5) = 2 \times 44.5 = 89 m$

Page No 781:

Question 5:

A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3. The area of the lawn is 3375 m². Find the cost of fencing the lawn at Rs 65 per metre.

Answer:

Let the length and breadth of the rectangular lawn be 5x m and 3x m, respectively.

Given:
Area of the rectangular lawn = $3375 m^{2}$

$\Rightarrow 3375 = 5 x \times 3 x \Rightarrow 3375 = 15 x^{2} \Rightarrow \frac{3375}{15} = x^{2} \Rightarrow 225 = x^{2} \Rightarrow x = 15$

Thus, we have:

$l = 5 x = 5 \times 15 = 75 m b = 3 x = 3 \times 15 = 45 m$

$Perimeter of the rectangular lawn = 2 (l + b) = 2 (75 + 45) = 2 (120) = 240 m$

Cost of fencing 1 m lawn = Rs 65
∴ Cost of fencing 240 m lawn = $240 \times 65 = Rs 15, 600$

Page No 781:

Question 6:

A room is 16 m long and 13.5 m broad. Find the cost of covering its floor with 75-m-wide carpet at ₹60 per metre.

Answer:

$As, the area of the floor = length \times breadth = 16 \times 13.5 = 216 m^{2} And, the width of the carpet = 75 m So, the length of the carpet required = \frac{Area of the floor}{Width of the carpet} = \frac{216}{75} = 2.88 m Now, the cost of the carpet required = 2.88 \times 60 = ₹ 172.80$

Hence, the cost of covering the floor with carpet is ₹172.80.

Disclaimer: The answer given in the textbook is incorrect. The same has been rectified above.

Page No 781:

Question 7:

The floor of a rectangular hall is 24 m and breadth 80 cm, will be required to cover the floor of the hall?

Answer:

Given:
Length = 24 m
Breadth = 18 m
Thus, we have:
$Area of the rectangular hall = 24 \times 18 = 432 m^{2}$

Length of each carpet = 2.5 m
Breadth of each carpet = 80 cm = 0.80 m

Area of one carpet = $2.5 \times 0.8 = 2 m^{2}$

Number of carpets required = $\frac{Area of the hall}{Area of the carpet} = \frac{432}{2} = 216$

Therefore, 216 carpets will be required to cover the floor of the hall.

Page No 781:

Question 8:

A 36 m-long, 15 broad verandah is to be paved with stones, each measuring 6 dm by 5 cm. How many stones will be required?

Answer:

$Area of the verandah = Length \times Breadth = 36 \times 15 = 540 m^{2}$
Length of the stone = 6 dm = 0.6 m
Breadth of the stone = 5 dm = 0.5 m
$Area of one stone = 0.6 \times 0.5 = 0.3 m^{2}$

$Number of stones r equired = \frac{Area of the verendah}{Area of the stone} = \frac{540}{0.3} = 1800$

Thus, 1800 stones will be required to pave the verandah.

Page No 781:

Question 9:

The area of a rectangle is 192 cm² and its perimeter is 56 cm. Find the dimensions of the rectangle.

Answer:

Area of the rectangle = 192 cm²
Perimeter of the rectangle = 56 cm

$Perimeter = 2 (length + breadth) \Rightarrow 56 = 2 (l + b) \Rightarrow l + b = 28 \Rightarrow l = 28 - b$

$Area = length \times breadth \Rightarrow 192 = (28 - b) x b \Rightarrow 192 = 28 b - b^{2} \Rightarrow b^{2} - 28 b + 192 = 0 \Rightarrow (b - 16) (b - 12) = 0 \Rightarrow b = 16 or 12$

Thus, we have;
$l = 28 - b \Rightarrow l = 28 - 12 \Rightarrow l = 16$

We will take length as 16 cm and breath as 12 cm because length is greater than breadth by convention.

Page No 781:

Question 10:

A rectangular park 35 m long 18 m wide is to be covered with grass, leaving 2.5 m uncovered all around it. Find the area to be laid with grass.

Answer:

The field is planted with grass, with 2.5 m uncovered on its sides.

The field is shown in the given figure.

Thus, we have;
Length of the area planted with grass = $35 - (2.5 + 2.5) = 35 - 5 = 30 m$

Width of the area planted with grass = $18 - (2.5 + 2.5) = 18 - 5 = 13 m$

Area of the rectangular region planted with grass = $30 \times 13 = 390 m^{2}$

Page No 781:

Question 11:

A rectangular plot measures 125 m by 78 m. It has gravel path 3 m wide all around on the outside. Find the area of the path and the cost of gravelling it at Rs 75 per m².

Answer:

The plot with the gravel path is shown in the figure.

Area of the rectangular plot = $l \times b$
Area of the rectangular plot = $125 \times 78 = 9750 m^{2}$
Length of the park including the path = 125 + 6 = 131 m
Breadth of the park including the path = 78 + 6 = 84 m
Area of the plot including the path
$= 131 \times 84 = 11004 m^{2}$

Area of the path = $11004 - 9750$
= 1254 m²
Cost of gravelling 1 m²of the path = Rs 75
∴ Cost of gravelling 1254 m²of the path = $1254 \times 75$
= Rs 94050

Page No 781:

Question 12:

A footpath of uniform width runs all around the inside of a rectangular field 54 m long and 35 m wide. If the area of the path is 420 m²., find the width of the path.

Answer:

Area of the rectangular field = $54 \times 35 = 1890 m^{2}$

Let the width of the path be x m. The path is shown in the following diagram:

Length of the park excluding the path = (54 $-$ 2x) m
Breadth of the park excluding the path = (35 $-$ 2x ) m

Thus, we have:
Area of the path = 420 m²
$\Rightarrow 420 = 54 \times 35 - (54 - 2 x) (35 - 2 x) \Rightarrow 420 = 1890 - (1890 - 70 x - 108 x + 4 x^{2}) \Rightarrow 420 = - 4 x^{2} + 178 x \Rightarrow 4 x^{2} - 178 x + 420 = 0 \Rightarrow 2 x^{2} - 89 x + 210 = 0 \Rightarrow 2 x^{2} - 84 x - 5 x + 210 = 0 \Rightarrow 2 x (x - 42) - 5 (x - 42) = 0 \Rightarrow (x - 42) (2 x - 5) = 0 \Rightarrow x - 42 = 0 or 2 x - 5 = 0 \Rightarrow x = 42 or x = 2.5$

The width of the path cannot be more than the breadth of the rectangular field.
∴ x = 2.5 m

Thus, the path is 2.5 m wide.

Page No 781:

Question 13:

The length and the breadth of a rectangular garden are in the ratio 9 : 5. A path 3.5 m wide, running all around inside it has an area of 1911 m². Find the dimensions of the garden.

Answer:

Let the length and breadth of the garden be 9x m and 5x m, respectively,
Now,
Area of the garden = $(9 x \times 5 x) = 45 x^{2}$
Length of the garden excluding the path = ( $9 x - 7)$
Breadth of the garden excluding the path = $(5 x - 7)$
Area of the path = $45 x^{2} - [(9 x - 7) (5 x - 7)]$
$\Rightarrow 1911 = 45 x^{2} - [45 x^{2} - 63 x - 35 x + 49] \Rightarrow 1911 = 45 x^{2} - 45 x^{2} + 63 x + 35 x - 49 \Rightarrow 1911 = 98 x - 49 \Rightarrow 1960 = 98 x \Rightarrow x = \frac{1960}{98} \Rightarrow x = 20$
Thus, we have:
Length = $9 x = 20 \times 9 = 180 m$
Breadth = $5 x = 5 \times 20 = 100 m$

Page No 781:

Question 14:

A room 4.9 m long and 3.5 m broad is covered with carpet, leaving an uncovered margin of 25 cm all around the room. If the breadth of the carpet is 80 cm, find its cost at Rs 80 per meter.

Answer:

Width of the room left uncovered = 0.25 m
Now,
Length of the room to be carpeted = $4.9 - (0.25 + 0.25) = 4.9 - 0.5 = 4.4 m$
Breadth of the room be carpeted = $3.5 - (0.25 + 0.25) = 3.5 - 0.5 = 3 m$

Area to be carpeted = $4.4 \times 3 = 13.2 m^{2}$

Breadth of the carpet = 80 cm = 0.8 m
We know:
Area of the room = Area of the carpet

$Length of the carpet = \frac{Area of the room}{Breadth of the carpet} = \frac{13.2}{0.8} = 16.5 m$

Cost of 1 m carpet = Rs 80
Cost of 16.5 m carpet = $80 \times 16.5 = Rs 1, 320$

Page No 781:

Question 15:

A carpet is laid on the floor of a room 8 m by 5 m. There is a border of constant width all around the carpet. If the area of the border is 12 m², find its width.

Answer:

Let the width of the border be x m.
The length and breadth of the carpet are 8 m and 5 m, respectively.
Area of the carpet = $8 \times 5 = 40 m^{2}$
Length of the carpet without border = $(8 - 2 x)$
Breadth of carpet without border = $(5 - 2 x)$
Area of the border = 12 m²
Area of the carpet without border = $(8 - 2 x) (5 - 2 x)$
$Thus, we have : 12 = 40 - [(8 - 2 x) (5 - 2 x)] \Rightarrow 12 = 40 - (40 - 26 x + 4 x^{2}) \Rightarrow 12 = 26 x - 4 x^{2}$

$\Rightarrow 26 x - 4 x^{2} = 12 \Rightarrow 4 x^{2} - 26 x + 12 = 0 \Rightarrow 2 x^{2} - 13 x + 6 = 0$

$\Rightarrow (2 x - 1) (x - 6) = 0 \Rightarrow 2 x - 1 = 0 and x - 6 = 0 \Rightarrow x = \frac{1}{2} and x = 6$

Because the border cannot be wider than the entire carpet, the width of the carpet is $\frac{1}{2}$ m, i.e., 50 cm.

Page No 781:

Question 16:

A 80 m by 64 m rectangular lawn has two roads, each 5 m wide, running through its middle, one parallel to its length and the other parallel to its breadth. Find the cost of gravelling the roads at Rs 40 per m².

Answer:

The length and breadth of the lawn are 80 m and 64 m, respectively.
The layout of the roads is shown in the figure below:

Area of the road ABCD = $80 \times 5 = 400 m^{2}$
Area of the road PQRS = $64 \times 5 = 320 m^{2}$
Clearly, the area EFGH is common in both the roads.
Area EFGH = $5 \times 5 = 25 m^{2}$
Area of the roads = $400 + 320 - 25$
= $695 m^{2}$
Given:
Cost of gravelling 1 m² area = Rs 40
∴ Cost of gravelling 695 m² area = $695 \times 40$
= Rs 27,800

Page No 781:

Question 17:

The dimensions of a room are 14 m × 10 m × 6.5 m. There are two doors and 4 windows in the room. Each door measures 2.5 m × 1.2 m and each window measures 1.5 × 1 m. Find the cost of painting the four walls of the room at Rs 35 per m².

Answer:

The room has four walls to be painted.

$Area of these walls = 2 (l \times h) + 2 (b \times h) = (2 \times 14 \times 6.5) + (2 \times 10 \times 6.5) = 312 m^{2}$

Now,
Area of the two doors = $(2 \times 2.5 \times 1.2) = 6 m^{2}$

Area of the four windows = $(4 \times 1.5 \times 1) = 6 m^{2}$

The walls have to be painted; the doors and windows are not to be painted.

∴ Total area to be painted $= 312 - (6 + 6) = 300 m^{2}$
Cost for painting 1 m² = Rs 35
Cost for painting 300 m² = $300 \times 35 = Rs 10, 500$

Page No 782:

Question 18:

The cost of painting the four walls of a room 12 m long at ₹30 per m² is ₹7560 and the cost of covering the floor with mat at ₹25 per m² is ₹2700. Find the dimensions of the room.

Answer:

$As, the rate of covering the floor = ₹ 25 per m^{2} And, the cost of covering the floor = ₹ 2700 So, the area of the floor = \frac{2700}{25} \Rightarrow length \times breadth = 108 \Rightarrow 12 \times breadth = 108 \Rightarrow breadth = \frac{108}{12} ∴ breadth = 9 m Also, As, the rate of painting the four walls = ₹ 30 per m^{2} And, the cost of painting the four walls = ₹ 7560 So, the area of the four walls = \frac{7560}{30} \Rightarrow 2 (length + breadth) height = 252 \Rightarrow 2 (12 + 9) height = 252 \Rightarrow 2 (21) height = 252 \Rightarrow 42 \times height = 252 \Rightarrow height = \frac{252}{42} ∴ height = 6 m$

So, the dimensions of the room are $12 m \times 9 m \times 6 m$ .

Page No 782:

Question 19:

Find the area and perimeter of a square plot of land whose diagonal is 24 m long. $[Take \sqrt{2} = 1.41]$

Answer:

Area of the square = $\frac{1}{2} \times {Diagonal}^{2}$

       = $\frac{1}{2} \times 24 \times 24$
             = $288 m^{2}$
Now, let the side of the square be x m.
Thus, we have:
$Area = {Side}^{2} \Rightarrow 288 = x^{2} \Rightarrow x = 12 \sqrt{2} \Rightarrow x = 16.92$

Perimeter = $4 \times Side$
   $= 4 \times 16.92 = 67.68 m$

Thus, the perimeter of the square plot is 67.68 m.

Page No 782:

Question 20:

Find the length of the diagonal of a square of area 128 cm². Also find the perimeter of the square, correct to two decimal places.

Answer:

Area of the square = 128 cm²
$Area = \frac{1}{2} d^{2} (where d is a diagonal of the square) \Rightarrow 128 = \frac{1}{2} d^{2} \Rightarrow d^{2} = 256 \Rightarrow d = 16 cm$

Now,
Area = ${Side}^{2}$
$\Rightarrow 128 = {Side}^{2} \Rightarrow Side = 11.31 cm$

Perimeter = 4(Side)
$= 4 (11.31) = 45.24 cm$

Page No 782:

Question 22:

The cost of harvesting a square field at ₹900 per hectare is ₹8100. Find the cost of putting a fence around it at ₹18 per metre.

Answer:

$As, the rate of the harvesting = ₹ 900 per hectare And, the cost of harvesting = ₹ 8100 So, the area of the square field = \frac{8100}{900} = 9 hectare \Rightarrow the area = 90000 m^{2} (As, 1 hectare = 10000 m^{2}) \Rightarrow {(side)}^{2} = 90000 \Rightarrow side = \sqrt{90000} So, side = 300 m Now, perimeter of the field = 4 \times side = 4 \times 300 = 1200 m Since, the rate of putting the fence = ₹ 18 per m So, the cost of putting the fence = 1200 \times 18 = ₹ 21, 600$

Page No 782:

Question 23:

The cost of fencing a square lawn at Rs 14 per metre is Rs 28000. Find the cost of mowing the lawn at Rs 54 per 100 m².

Answer:

Cost of fencing the lawn = Rs 28000
Let l be the length of each side of the lawn. Then, the perimeter is 4l.
We know:
$Cost = Rate \times Perimeter \Rightarrow 28000 = 14 \times 4 l$

$\Rightarrow 28000 = 56 l O r, l = \frac{28000}{56} = 500 m$

Area of the square lawn $= 500 \times 500 = 250000 m^{2}$

Cost of mowing 100 m² of the lawn = Rs 54
Cost of mowing 1 m² of the lawn $= Rs \frac{54}{100}$

∴ Cost of mowing 250000 m² of the lawn $= \frac{250000 \times 54}{100} = Rs 135000$

Page No 782:

Question 24:

In the given figure, ABCD is a quadrilateral in which diagonal BD = 24 cm, AL $⊥$ BD and CM $⊥$ BD such that AL = 9 cm and CM = 12 cm. Calculate the area of the quadrilateral.

Answer:

$We have, BD = 24 cm, AL = 9 cm, CM = 12 cm, AL ⊥ BD and CM ⊥ BD Area of the quadrilateral = ar (∆ ABD) + ar (∆ BCD) = \frac{1}{2} \times BD \times AL + \frac{1}{2} \times BD \times CM = \frac{1}{2} \times 24 \times 9 + \frac{1}{2} \times 24 \times 12 = 108 + 144 = 252 {cm}^{2}$

So, the area of the quadrilateral ABCD is 252 cm².

Page No 782:

Question 25:

Find the area of the quadrilateral ABCD in which AD = 24 cm, ∠BAD = 90° and BCD forms an equilateral triangle whose each side is equal to 26 cm. Also find the perimeter of the quadrilateral. Take $\sqrt{3}$ = 1.73.

Answer:

$∆$ BDC is an equilateral triangle with side a= 26 cm.
Area of $∆ B D C = \frac{\sqrt{3}}{4} a^{2}$
$= \frac{\sqrt{3}}{4} \times 26^{2} = \frac{1.73}{4} \times 676 = 292.37 {cm}^{2}$

By using Pythagoras' theorem in the right-angled triangle $∆$ DAB, we get:
$A D^{2} + A B^{2} = B D^{2} \Rightarrow 24^{2} + A B^{2} = 26^{2} \Rightarrow A B^{2} = 26^{2} - 24^{2} \Rightarrow A B^{2} = 676 - 576 \Rightarrow A B^{2} = 100 \Rightarrow A B = 10 cm$

Area of $∆ A B D = \frac{1}{2} \times b \times h$
$= \frac{1}{2} \times 10 \times 24 = 120 {cm}^{2}$

Area of the quadrilateral = Area of $∆$ BCD + Area of $∆$ ABD
   $= 292.37 + 120$
           = 412.37 cm²

Perimeter of the quadrilateral = AB + BC + CD + AD
                                                 = 24 + 10 + 26 + 26
                                                 = 86 cm

Page No 782:

Question 26:

Find the perimeter and area of the quadrilateral ABCD in which AB = 17 cm, AD = 9 cm, CD = 12 cm, ∠ACB = 90° and AC = 15 cm.

Answer:

In the right-angled $∆$ ACB:

$A B^{2} = B C^{2} + A C^{2} \Rightarrow 17^{2} = B C^{2} + 15^{2} \Rightarrow 17^{2} - 15^{2} = B C^{2} \Rightarrow 64 = B C^{2} \Rightarrow B C = 8 cm$

Perimeter = $A B + B C + C D + A D$
     = $17 + 8 + 12 + 9$
   = 46 cm

Area of $∆ A B C = \frac{1}{2} (b \times h)$

   = $\frac{1}{2} (8 \times 15)$
= $60 {cm}^{2}$
$In ∆ A D C : A C^{2} = A D^{2} + C D^{2} So, ∆ A D C is a right - angled triangle at D .$

Area of $∆ A D C = \frac{1}{2} \times b \times h$
    $= \frac{1}{2} \times 9 \times 12 = 54 {cm}^{2}$

∴ Area of the quadrilateral = Area of $∆$ ABC + Area of $∆$ ADC
   = 60 + 54
   = 114 cm²

Page No 782:

Question 27:

Find the area of the quadrilateral ABCD in which AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm and diagonal BD = 20 cm.

Answer:

Area of $∆$ ABD = $\sqrt{s (s - a) (s - b) (s - c)}$
$s = \frac{1}{2} (a + b + c) s = \frac{42 + 20 + 34}{2} s = 48 cm$

Area of $∆$ ABD = $\sqrt{48 (48 - 42) (48 - 20) (48 - 34)}$

      $= \sqrt{48 \times 6 \times 28 \times 14} = \sqrt{112896} = 336 {cm}^{2}$

Area of $∆$ BDC = $\sqrt{s (s - a) (s - b) (s - c)}$
$s = \frac{1}{2} (a + b + c) s = \frac{21 + 20 + 29}{2} s = 35 cm$

Area of $∆$ BDC = $\sqrt{35 (35 - 29) (35 - 20) (35 - 21)}$

$= \sqrt{35 \times 6 \times 15 \times 14} = \sqrt{44100} = 210 {cm}^{2}$

∴ Area of quadrilateral ABCD = Area of $∆$ ABD + Area of $∆$ BDC
       = 336 + 210
        = 546 cm²

Page No 782:

Question 28:

Find the area of a parallelogram with base equal to 25 cm and the corresponding height measuring 16.8 cm.

Answer:

Given:
Base = 25 cm
Height = 16.8 cm
∴ Area of the parallelogram $= Base \times Height = 25 cm \times 16.8 cm = 420 {cm}^{2}$

Page No 783:

Question 29:

The adjacent sides of a parallelogram are 32 cm and 24 cm. If the distance between the longer sides is 17.4 cm, find hte distance between the shorter sides.

Answer:

Longer side = 32 cm
Shorter side = 24 cm
Let the distance between the shorter sides be x cm.
Area of a parallelogram = $Longer side \times Distance between the longer sides$
= $Shorter side \times Distance between the shorter sides$
or, $32 \times 17.4 = 24 \times x$

or, $x = \frac{32 \times 17.4}{24} = 23.2 cm$

∴ Distance between the shorter sides = 23.2 cm

Page No 783:

Question 30:

The area of a parallelogram is 392 m². If its altitude is twice the corresponding base, determine the base and the altitude.

Answer:

Area of the parallelogram = 392 m²
Let the base of the parallelogram be b m.
Given:
Height of the parallelogram is twice the base.
∴ Height = 2b m
Area of a parallelogram = $Base x Height$
$\Rightarrow 392 = b \times 2 b \Rightarrow 392 = 2 b^{2} \Rightarrow \frac{392}{2} = b^{2} \Rightarrow 196 = b^{2} \Rightarrow b = 14$
∴ Base = 14 m
Altitude = $2 \times Base = 2 \times 14 = 28 m$

Page No 783:

Question 31:

The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm, and the diagonal AC measures 42 cm. Find the area of the parallelogram.

Answer:

Parallelogram ABCD is made up of congruent $∆$ ABC and $∆$ ADC.

Area of triangle ABC = $\sqrt{s (s - a) (s - b) (s - c)}$ (Here, s is the semiperimeter.)

Thus, we have:

$s = \frac{a + b + c}{2} s = \frac{34 + 20 + 42}{2} s = 48 cm$

$Area of ∆ A B C = \sqrt{48 (48 - 34) (48 - 20) (48 - 42)} = \sqrt{48 \times 14 \times 28 \times 6} = 336 {cm}^{2}$

Now,
Area of the parallelogram = $2 \times Area of ∆ A B C$
$= 2 \times 336 = 672 {cm}^{2}$

Page No 783:

Question 32:

Find the area of the rhombus, the lengths of whose diagonals are 30 cm and 16 cm. Also find the perimeter of the rhombus.

Answer:

Area of the rhombus = $\frac{1}{2} \times d_{1 \times} d_{2}$ , where d₁ and d₂ are the lengths of the diagonals.

      $= \frac{1}{2} \times 30 \times 16 = 240 {cm}^{2}$

Side of the rhombus = $\frac{1}{2} \sqrt{{d_{1}}^{2} + {d_{2}}^{2}}$

   $= \frac{1}{2} \sqrt{30^{2} + 16^{2}} = \frac{1}{2} \sqrt{1156} = \frac{1}{2} \times 34 = 17 cm$

Perimeter of the rhombus = $4 a$
      = $4 \times 17$
= 68 cm

Page No 783:

Question 33:

The perimeter of a rhombus is 60 cm. If one of its diagonals is 18 cm long, find (i) the length of the other diagonal, and (ii) the area of the rhombus.

Answer:

Perimeter of a rhombus = 4a (Here, a is the side of the rhombus.)

$\Rightarrow 60 = 4 a \Rightarrow a = 15 cm$

(i) Given:
One of the diagonals is 18 cm long.
$d_{1} = 18 cm$

Thus, we have:
$Side = \frac{1}{2} \sqrt{{d_{1}}^{2} + {d_{2}}^{2}} \Rightarrow 15 = \frac{1}{2} \sqrt{18^{2} + {d_{2}}^{2}} \Rightarrow 30 = \sqrt{18^{2} + {d_{2}}^{2}} Squaring both sides, we get : \Rightarrow 900 = 18^{2} + {d_{2}}^{2} \Rightarrow 900 = 324 + {d_{2}}^{2} \Rightarrow {d_{2}}^{2} = 576 \Rightarrow d_{2} = 24 cm$

∴ Length of the other diagonal = 24 cm

(ii) Area of the rhombus $= \frac{1}{2} d_{1} \times d_{2}$
$= \frac{1}{2} \times 18 \times 24 = 216 {cm}^{2}$

Page No 783:

Question 34:

The area of a rhombus is 480 cm², and one of its diagonals measures 48 cm. Find (i) the length of the other diagonal, (ii) the length of each of its sides, and (iii) its perimeter.

Answer:

i) Area of a rhombus $= \frac{1}{2} \times d_{1} \times d_{2}$ , where d₁ and d₂ are the lengths of the diagonals.
$\Rightarrow 480 = \frac{1}{2} \times 48 \times d_{2} \Rightarrow d_{2} = \frac{480 \times 2}{48} \Rightarrow d_{2} = 20 cm$

∴ Length of the other diagonal = 20 cm

(ii) Side = $\frac{1}{2} \sqrt{{d_{1}}^{2} + {d_{2}}^{2}}$

$= \frac{1}{2} \sqrt{48^{2} + 20^{2}} = \frac{1}{2} \sqrt{2304 + 400} = \frac{1}{2} \sqrt{2704} = \frac{1}{2} \times 52 = 26 cm$

∴ Length of the side of the rhombus = 26 cm

(iii) Perimeter of the rhombus = $4 \times Side$
$= 4 \times 26 = 104 cm$

Page No 783:

Question 35:

The parallel sides of a trapezium are 12 cm and 9 cm and the distance between them is 8 cm. Find the area of the trapezium.

Answer:

$Area of the trapezium = \frac{1}{2} \times (sum of the parallel sides) \times distance between the parallel sides = \frac{1}{2} \times (12 + 9) \times 8 = 21 \times 4 = 84 {cm}^{2}$

So, the area of the trapezium is 84 cm².

Page No 783:

Question 36:

The shape of the cross section of a canal is a trapezium. If the canal is 10 m wide at the top, 6 m wide at the bottom and the area of its cross section is 640 m², find the depth of the canal.

Answer:

Area of the canal = $640 m^{2}$
Area of trapezium = $\frac{1}{2} \times (Sum of parallel sides) \times (Distance between them)$
$\Rightarrow 640 = \frac{1}{2} \times (10 + 6) \times h \Rightarrow \frac{1280}{16} = h \Rightarrow h = 80 m$

Therefore, the depth of the canal is 80 m.

Page No 783:

Question 37:

Find the area of a trapezium whose parallel sides are 11 ma and 25 m long, and the nonparallel sides are 15 m and 13 m long.

Answer:

Draw $D E ∥ B C$ and DL perpendicular to AB.
The opposite sides of quadrilateral DEBC are parallel. Hence, DEBC is a parallelogram.
∴ DE = BC = 13 m
Also,
$A E = (A B - E B) = (A B - D C) = (25 - 11) = 14 m$
For $∆ D A E$ :
Let:
AE = a =14 m
DE = b = 13 m
DA = c =15 m

Thus, we have:

$s = \frac{a + b + c}{2} s = \frac{14 + 13 + 15}{2} = 21 m$

Area of $∆ D A E = \sqrt{s (s - a) (s - b) (s - c)}$
$= \sqrt{21 \times (21 - 14) \times (21 - 13) \times (21 - 15)} = \sqrt{21 \times 7 \times 8 \times 6} = \sqrt{7056} = 84 m^{2}$

Area of $∆ D A E = \frac{1}{2} \times A E \times D L$
$\Rightarrow 84 = \frac{1}{2} \times 14 \times D L \Rightarrow \frac{84 \times 2}{14} = D L \Rightarrow D L = 12 m$

Area of trapezium = $\frac{1}{2} \times (Sum of parallel sides) \times (Distance between them)$
$= \frac{1}{2} \times (11 + 25) \times 12 = \frac{1}{2} \times 36 \times 12 = 216 m^{2}$

Page No 786:

Question 1:

The length of a rectangular hall is 5 m more than its breadth. If the area of the hall is 750 m², then its length is
(a) 15 m
(b) 20 m
(c) 25 m
(d) 30 m

Answer:

(d) 30 m

Let the length of the rectangle be x m.
∴ Breadth of the rectangle $= (x - 5) m$

$Area = x (x - 5) = x^{2} - 5 x \Rightarrow x^{2} - 5 x = 750$
$\Rightarrow x^{2} - 5 x - 750 = 0 \Rightarrow x^{2} - 30 x + 25 x - 750 = 0 \Rightarrow x (x - 30) + 25 (x - 30) = 0 \Rightarrow (x + 25) (x - 30) = 0 \Rightarrow x + 25 = 0 and x - 30 = 0 \Rightarrow x = - 25 and x = 30$

Length cannot be negative.
∴ Length = x = 30 m

Page No 786:

Question 2:

The length of a rectangular field is 23 m more than its breadth. If the perimeter of the field is 206 m, then its area is
(a) 2420 m²
(b) 2520 m²
(c) 2480 m²
(d) 2620 m²

Answer:

(b) 2520 m²

Let the breadth of the field be x m.
∴ Length = (x + 23) m

Now,
Perimeter $= 2 (Length + Breadth) = 2 (x + x + 23) = (4 x + 46) m$
Thus, we have:
$4 x + 46 = 206 \Rightarrow 4 x = 206 - 46 = 160 \Rightarrow x = \frac{160}{4} = 40$
∴ Breadth = x = 40 m
Length = x + 23 $= 40 + 23 = 63 m$
Area $= Length \times Breadth = 63 \times 40 = 2520 m^{2}$

Page No 786:

Question 3:

The length of a rectangular field is 12 m and the length of its diagonal is 15 m. The area of the field is
(a) 108 m²
(b) 180 m²
(c) $30 \sqrt{3} m^{2}$
(d) $12 \sqrt{15} m^{2}$

Answer:

(a) 108 m²

Length of the rectangular field = 12 m
Diagonal = 15 m

${Diagonal}^{2} = {Length}^{2} + {Breadth}^{2}$
$Breadth = \sqrt{{Diagonal}^{2} - {Length}^{2}} = \sqrt{15^{2} - 12^{2}} = \sqrt{225 - 144} = 9 m$

∴ Area of the field $= Length \times Breadth = 12 \times 9 = 108 m^{2}$

Page No 786:

Question 4:

Choose the correct answer of the following question:

The cost of carpeting a room 15 m long with a carpet 75 cm wide, at ₹70 per metre, is ₹8400. The width of the room is

(a) 9 m (b) 8 m (c) 6 m (d) 12 m

Answer:

$We have, Width of the carpet = 75 cm = 0.75 m and length of the room = 15 m Length of the carpet = \frac{Cost of carpeting}{Rate of carpeting} = \frac{8400}{70} = 120 m Now, area of the carpet required for carpeting = 120 \times 0.75 m^{2} \Rightarrow area of the floor = 120 \times 0.75 m^{2} \Rightarrow 15 \times breadth = 120 \times 0.75 \Rightarrow breadth = \frac{120 \times 0.75}{15} So, breadth = 6 m$

Hence, the correct answer is option (c).

Page No 786:

Question 5:

The length of a rectangle is thrice its breadth and the length of its diagonal is $8 \sqrt{10} cm .$ . The perimeter of the rectangle is
(a) $15 \sqrt{10} cm$
(b) $16 \sqrt{10} cm$
(c) $24 \sqrt{10} cm$
(d) 64 cm

Answer:

(d) 64 cm

Let the breadth of the rectangle be x cm.
∴ Length of the rectangle = 3x cm
We know:
$Diagonal = \sqrt{(Length)^{2} + (Breadth)^{2}} \Rightarrow 8 \sqrt{10} = \sqrt{x^{2} + (3 x)^{2}} \Rightarrow 8 \sqrt{10} = \sqrt{x^{2} + 9 x^{2}} \Rightarrow 8 \sqrt{10} = x \sqrt{10} \Rightarrow x = 8$

Now,
Breadth of the rectangle = x = 8 cm
Length of the rectangle = 3x = 24 cm
Perimeter of the rectangle $= 2 (Length + Breadth) = 2 (8 + 24) = 64 cm$

Page No 786:

Question 6:

On increasing the length of a rectangle by 20% and decreasing its breadth by 20%, what is the change in its area?
(a) 20% increase
(b) 20% decrease
(c) No change
(d) 4% decrease

Answer:

(d) 4% decrease

Let:
Length $= x$
breadth $= y$
Area $= x y$

Now,
New length $= x + 20 % x = x + \frac{1}{5} x = \frac{6}{5} x$

New breadth $= y - 20 % y = y - \frac{1}{5} y = \frac{4}{5} y$

New area $= \frac{6}{5} x \times \frac{4}{5} y = \frac{24}{25} x y$

Difference in the areas $= x y - \frac{24}{25} x y = \frac{1}{25} x y$

Difference in percentage $= [(\frac{\frac{1}{25} x y}{x y}) \times 100] % = 4 %$

Page No 786:

Question 7:

A rectangular ground 80 m × 50 m has a path 1 m wide outside around it. The area of the path is
(a) 264 m²
(b) 284 m²
(c) 400 m²
(d) 464 m²

Answer:

(a) 264 m²

Length of the ground including the path $= 80 + 2 = 82 m$
Breadth of the ground including the path $= 50 + 2 = 52 m$

Total area (including the path) $= Length \times Breadth = 82 \times 52 = 4264 m^{2}$

Area of the field $= 80 \times 50 = 4000 m^{2}$

Area of the path $= 4264 - 4000 = 264 m^{2}$

Page No 786:

Question 8:

The length of the diagonal of a square is $10 \sqrt{2} cm .$ Its area is
(a) 200 cm²
(b) 100 cm²
(c) 150 cm²
(d) 100 $\sqrt{2}$ cm²

Answer:

(b) 100 cm²

A diagonal of a square forms the hypotenuse of a right-angled triangle with base and height equal to side a.

${Diagonal}^{2} = a^{2} + a^{2} \Rightarrow {Diagonal}^{2} = 2 a^{2} \Rightarrow a = \frac{1}{\sqrt{2}} Diagonal = \frac{1}{\sqrt{2}} \times 10 \sqrt{2} = 10 cm$

∴ Area of the square $= a^{2} = 10 \times 10 = 100 {cm}^{2}$

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Question 9:

The area of a square field is 6050 m². The length of its diagonal is
(a) 135 m
(b) 120 m
(c) 112 m
(d) 110 m

Answer:

(d) 110 m

Let the diagonal of the square field be d m.

In case of a square field, $d^{2} = 2 a^{2}$ , where a is the side of the square field.
Now,
$Area of a square field = a^{2}$
$d^{2} = 2 a^{2} \Rightarrow d^{2} = 2 \times A r e a o f t h e s q u a r e f i e l d \Rightarrow d = \sqrt{2 \times Area of the square field}$

∴ $d = \sqrt{2 \times 6050} = \sqrt{12100} = 110$

Page No 786:

Question 10:

The length of a square field is 0.5 hectare. The length of its diagonal is
(a) 150 m
(b) $100 \sqrt{2}$
(c) 100 m
(d) $50 \sqrt{2} m$

Answer:

(c) 100 m

Disclaimer :- The length cannot be in hectare So we used is as area of the square.
Area of the square field $= 0.5 \times 10000 = 5000 m^{2}$

The diagonal divides the square into two isosceles right-angled triangles.

Using Pythagoras' theorem, we have:

${Diagonal}^{2} = a^{2} + a^{2} = 2 a^{2}$
Area of a square = $a^{2}$

∴ Diagonal $= \sqrt{2 area} = \sqrt{2 \times 5000} = \sqrt{10000} = 100 m$

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Question 11:

The area of an equilateral triangle is $4 \sqrt{3} {cm}^{2}$ . Its perimeter is
(a) 9 cm
(b) 12 cm
(c) $12 \sqrt{3}$ cm
(d) $6 \sqrt{3}$ cm

Answer:

(b) 12 cm

Area of an equilateral triangle $= \frac{\sqrt{3}}{4} a^{2}$ (where a is the length of the side)
Thus, we have:
$4 \sqrt{3} = \frac{\sqrt{3}}{4} a^{2}$
$\Rightarrow a^{2} = 16 \Rightarrow a = 4 cm$

Perimeter of the equilateral triangle = 3a $= 3 \times 4 = 12 cm$

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Question 12:

Each side of an equilateral triangle is 8 cm. Its area is
(a) 24 cm²
(b) $24 \sqrt{3} {cm}^{2}$
(c) $16 \sqrt{3} {cm}^{2}$
(d) $8 \sqrt{3} {cm}^{2}$

Answer:

(c) $16 \sqrt{3} {cm}^{2}$

Let the side of the equilateral triangle be a.
Given:
a = 8 cm
Now,
Area of the equilateral triangle $= \frac{\sqrt{3}}{4} a^{2} = \frac{\sqrt{3}}{4} \times 8 \times 8 = 16 \sqrt{3} {cm}^{2}$

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Question 13:

Choose the correct answer of the following question:

Each side of an equilateral triangle is $6 \sqrt{3}$ cm. The altitude of the triangle is

(a) 8 cm (b) 9 cm (c) $3 \sqrt{3}$ cm (d) 6 cm

Answer:

$As, area of an equilateral triangle = \frac{\sqrt{3}}{4} \times {(side)}^{2} \Rightarrow \frac{1}{2} \times Base \times Height = \frac{\sqrt{3}}{4} \times {(6 \sqrt{3})}^{2} \Rightarrow \frac{1}{2} \times 6 \sqrt{3} \times Height = \frac{\sqrt{3}}{4} \times 36 \times 3 \Rightarrow 3 \sqrt{3} \times Height = 27 \sqrt{3} \Rightarrow Height = \frac{27 \sqrt{3}}{3 \sqrt{3}} ∴ Height = 9 cm$

Hence, the correct option is (b).

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Question 14:

Choose the correct answer of the following question:

The height of an equilateral triangle is $3 \sqrt{3}$ cm. Its area is

(a) $6 \sqrt{3}$ cm² (b) 27 cm² (c) $9 \sqrt{3}$ cm² (d) $27 \sqrt{3}$ cm²

Answer:

$Let the side of the equilateral triangle be x . As, area of the equilateral triangle = \frac{\sqrt{3}}{4} \times {(side)}^{2} \Rightarrow \frac{1}{2} \times Base \times Height = \frac{\sqrt{3}}{4} \times x^{2} \Rightarrow \frac{1}{2} \times x \times 3 \sqrt{3} = \frac{x^{2} \sqrt{3}}{4} \Rightarrow \frac{3 x \sqrt{3}}{2} = \frac{x^{2} \sqrt{3}}{4} \Rightarrow \frac{3 \sqrt{3}}{2} = \frac{x \sqrt{3}}{4} \Rightarrow x = \frac{3 \sqrt{3} \times 4}{2 \sqrt{3}} \Rightarrow x = 6 cm Now, the area of the triangle = \frac{\sqrt{3}}{4} \times 6^{2} = \frac{\sqrt{3}}{4} \times 36 = 9 \sqrt{3} {cm}^{2}$

Hence, the correct answer is option (c).

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Question 15:

Choose the correct answer of the following question:

The base and height of a triangle are in the ratio 3:4 and its area is 216 cm². The height of the triangle is

(a) 18 cm (b) 24 cm (c) 21 cm (d) 28 cm

Answer:

$Let the base of the triangle be 3 x and its height be 4 x . As, the area of the triangle = 216 {cm}^{2} \Rightarrow \frac{1}{2} \times Base \times Height = 216 \Rightarrow \frac{1}{2} \times 3 x \times 4 x = 216 \Rightarrow 6 x^{2} = 216 \Rightarrow x^{2} = \frac{216}{6} \Rightarrow x^{2} = 36 \Rightarrow x = \sqrt{36} \Rightarrow x = 6 cm So, the height of the triangle = 4 \times 6 = 24 cm$

Hence, the correct answer is option (b).

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Question 16:

Choose the correct answer of the following question:

The lengths of the sides of a triangular field are 20 m, 21 m and 29 m. The cost of cultivating the field at ₹9 per m² is

(a) ₹2610 (b) ₹3780 (c) ₹1890 (d) ₹1800

Answer:

$As, the sides of the triangle are 20 m, 21 m and 29 m So, the semi - perimeter = \frac{20 + 21 + 29}{2} = 35 m Now, the area of the triangular field = \sqrt{35 (35 - 20) (35 - 21) (35 - 29)} = \sqrt{35 \times 15 \times 14 \times 6} = \sqrt{7 \times 5 \times 5 \times 3 \times 7 \times 2 \times 2 \times 3} = 2 \times 3 \times 5 \times 7 = 210 m^{2} ∴ The cost of cultivating the field = 210 \times 9 = ₹ 1890$

Hence, the correct answer is option (c).

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Question 17:

The side of a square is equal to the side of an equilateral triangle. The ratio of their areas is
(a) 4 : 3
(b) $2 : \sqrt{3}$
(c) $4 : \sqrt{3}$
(d) none of these

Answer:

(c) $4 : \sqrt{3}$

Let:
Length of the side of the square = Length of the side of the equilateral triangle = a unit

Now,

Area of the square $= a \times a = a^{2} {unit}^{2}$

Area of the equilateral triangle $= \frac{\sqrt{3}}{4} a^{2} {unit}^{2}$

$Ratio of areas = \frac{Area of the square}{Area of the equilateral triangle} = \frac{a^{2}}{\frac{\sqrt{3}}{4} a^{2}} = \frac{4}{\sqrt{3}}$

Page No 787:

Question 18:

The side of an equilateral triangle is equal to the radius of a circle whose area is 154 cm². The area of the triangle is
(a) 49 cm²
(b) $\frac{49 \sqrt{3}}{4} {cm}^{2}$
(c) $\frac{7 \sqrt{3}}{4} {cm}^{2}$
(d) 77 cm²

Answer:

(b) $\frac{49 \sqrt{3}}{4} {cm}^{2}$

Area of a circle $= π r^{2}$
$\Rightarrow 154 = π r^{2} \Rightarrow r = \sqrt{\frac{154 \times 7}{22}} = \sqrt{7 \times 7} = 7 cm$

The radius of the circle is equal to the side of the equilateral triangle.

∴ r = a (Here, a is the side of the equilateral triangle.)

$a = 7 cm$

∴ Area of the equilateral triangle $= \frac{\sqrt{3}}{4} a^{2} = \frac{\sqrt{3}}{4} \times 7 \times 7 = \frac{49 \sqrt{3}}{4} {cm}^{2}$

Page No 787:

Question 19:

Choose the correct answer of the following question:

The area of a rhombus is 480 cm² and the length of one of its diagonals is 20 cm. The length of each side of the rhombus is

(a) 24 cm (b) 30 cm (c) 26 cm (d) 28 cm

Answer:

$We have, BD = 20 cm \Rightarrow BO = \frac{BD}{2} = \frac{20}{2} = 10 cm As, area of the rhombus ABCD = 480 {cm}^{2} \Rightarrow \frac{1}{2} \times AC \times BD = 480 \Rightarrow \frac{1}{2} \times AC \times 20 = 480 \Rightarrow AC \times 10 = 480 \Rightarrow AC = \frac{480}{10} \Rightarrow AC = 48 cm \Rightarrow AO = \frac{AC}{2} = \frac{48}{2} = 24 cm Now, in ∆ AOB, Using Pythagoras theorem, {AB}^{2} = {AO}^{2} + {BO}^{2} = 24^{2} + 10^{2} = 576 + 100 \Rightarrow {AB}^{2} = 676 \Rightarrow AB = \sqrt{676} ∴ AB = 26 cm$

Hence, the correct answer is option (c).

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Question 20:

Choose the correct answer of the following question:

One side of a rhombus is 20 cm long and one of its diagonals measures 24 cm. The area of the rhombus is

(a) 192 cm² (b) 480 cm² (c) 240 cm² (d) 384 cm²

Answer:

$WE have, AB = BC = CD = DA = 20 cm and BD = 24 cm Also, BO = \frac{BD}{2} = \frac{24}{2} = 12 cm In ∆ AOB, Using Pythagoras theorem {AO}^{2} = {AB}^{2} - {BO}^{2} = 20^{2} - 12^{2} = 400 - 144 \Rightarrow {AO}^{2} = 256 \Rightarrow AO = \sqrt{256} \Rightarrow AO = 16 cm \Rightarrow AC = 2 AO = 2 \times 16 = 32 cm Now, the area of the rhombus ABCD = \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 32 \times 24 = 384 {cm}^{2}$

Hence, the correct answer is option (d).

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Question 1:

In the given figure ABCD is a quadrilateral in which ∠ABC = 90°, ∠BDC = 90°, AC = 17 cm, BC = 15 cm, BD = 12 cm and CD = 9 cm. The area of quad. ABCD is

(a) 102 cm²
(b) 114 cm²
(c) 95 cm²
(d) 57 cm²

Answer:

(b) 114 sq cm

Using Pythagoras' theorem in $∆$ ABC, we get:
$A C^{2} = A B^{2} + B C^{2} \Rightarrow A B = \sqrt{A C^{2} - B C^{2}} = \sqrt{17^{2} - 15^{2}} = 8 cm$

$Area of ∆ A B C = \frac{1}{2} \times A B \times B C = \frac{1}{2} \times 8 \times 15 = 60 {cm}^{2}$

$Area of ∆ B C D = \frac{1}{2} \times B D \times C D = \frac{1}{2} \times 12 \times 9 = 54 {cm}^{2}$

∴ Area of quadrilateral ABCD $= Ar (∆ A B C) + Ar (∆ B C D) = 54 + 60 = 114 {cm}^{2}$

Page No 790:

Question 2:

In the given figure ABCD is a trapezium in which AB = 40 m, BC = 15 m, CD = 28 m, AD = 9 m and CE ⊥ AB. Area of trap. ABCD is

(a) 306 m²
(b) 316 m²
(c) 296 m²
(d) 284 m²

Answer:

(a) 306 m²

In the given figure, AECD is a rectangle.

Length AE = Length CD = 28 m

Now,
$B E = A B - A E = 40 - 28 = 12 m$

Also,
AD = CE = 9 m
$Area of trapezium = \frac{1}{2} \times Sum of parallel sides \times Distance between them = \frac{1}{2} \times (D C + A B) \times C E = \frac{1}{2} \times (28 + 40) \times 9 = \frac{1}{2} \times 68 \times 9 = 306 m^{2}$

In the given figure, if DA is perpendicular to AE, then it can be solved, otherwise it cannot be solved.

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Question 3:

The sides of a triangle are in the ratio 12 : 14 : 25 and its perimeter is 25.5 cm. The largest side of the triangle is
(a) 7 cm
(b) 14 cm
(c) 12.5 cm
(d) 18 cm

Answer:

(c) 12.5 cm

Let the sides of the triangle be 12x cm, 14x cm and 25x cm.
Thus, we have:
$Perimeter = 12 x + 14 x + 25 x \Rightarrow 25.5 = 51 x \Rightarrow x = \frac{25.5}{51} = 0.5$

∴ Greatest side of the triangle $= 25 x = 25 \times 0.5 = 12.5 cm$

Page No 790:

Question 4:

The parallel sides of a trapezium are 9.7 cm and 6.3 cm, and the distance between them is 6.5 cm. The area of the trapezium is
(a) 104 cm²
(b) 78 cm²
(c) 52 cm²
(d) 65 cm²

Answer:

(c) 52 cm²

$Area of trapezium = \frac{1}{2} (Sum of parallel sides) \times Distance between them = \frac{1}{2} \times (9.7 + 6.3) \times 6.5 = 8 \times 6.5 = 52.0 {cm}^{2}$

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Question 5:

Find the area of an equilateral triangle having each side of length 10 cm.

Answer:

Given:
Side of the equilateral triangle = 10 cm
Thus, we have:
$Area of the equilateral triangle = \frac{\sqrt{3}}{4} {side}^{2} = \frac{\sqrt{3}}{4} \times 10 \times 10 = 25 \times 1.732 = 43.3 {cm}^{2}$

Page No 790:

Question 6:

Find the area of an isosceles triangle each of whose equal sides is 13 cm and whose base is 24 cm.

Answer:

Area of an isosceles triangle:
$= \frac{1}{4} b \sqrt{4 a^{2} - b^{2}} (Where a is the length of the equal sides and b is the base) = \frac{1}{4} \times 24 \sqrt{4 {(13)}^{2} - 24^{2}} = 6 \sqrt{4 \times 169 - 576} = 6 \sqrt{676 - 576} = 6 \sqrt{100} = 6 \times 10 = 60 {cm}^{2}$

Page No 790:

Question 7:

The longer side of a rectangular hall is 24 m and the length of its diagonal is 26 m. Find the area of the hall.

Answer:

Let the rectangle ABCD represent the hall.

Using the Pythagorean theorem in the right-angled triangle ABC, we have:
${Diagonal}^{2} = {Length}^{2} + {Breadth}^{2}$

$\Rightarrow Breadth = \sqrt{{Diagonal}^{2} - {Length}^{2}} = \sqrt{26^{2} - 24^{2}} = \sqrt{676 - 576} = \sqrt{100} = 10 m$

∴ Area of the hall $= Length \times Breadth = 24 \times 10 = 240 m^{2}$

Page No 790:

Question 8:

The length of the diagonal of a square is 24 cm. Find its area.

Answer:

The diagonal of a square forms the hypotenuse of an isosceles right triangle. The other two sides are the sides of the square of length a cm.

Using Pythagoras' theorem, we have:

${Diagonal}^{2} = a^{2} + a^{2} = 2 a^{2} \Rightarrow Diagonal = \sqrt{2} a$

$Diagonal of the square = \sqrt{2} a \Rightarrow 24 = \sqrt{2} a$

$\Rightarrow$ a $= \frac{24}{\sqrt{2}}$
Area of the square $= {Side}^{2} = {(\frac{24}{\sqrt{2}})}^{2} = \frac{24 \times 24}{2} = 288 {cm}^{2}$

Page No 790:

Question 9:

Find the area of a rhombus whose diagonals are 48 cm and 20 cm long.

Answer:

$Area of the rhombus = \frac{1}{2} (Product of diagonals) = \frac{1}{2} (48 \times 20) = 480 {cm}^{2}$

Page No 790:

Question 10:

Find the area of a triangle whose sides are 42 cm, 34 cm and 20 cm.

Answer:

To find the area of the triangle, we will first find the semiperimeter of the triangle.

Thus, we have:

$s = \frac{1}{2} (a + b + c) = \frac{1}{2} (42 + 34 + 20) = \frac{1}{2} \times 96 = 48 cm$

Now,

$Area of the triangle = \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{48 (48 - 42) (48 - 34) (48 - 20)} = \sqrt{48 \times 6 \times 14 \times 28} = \sqrt{112896} = 336 {cm}^{2}$

Page No 790:

Question 11:

A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3 and its area is 3375 m². Find the cost of fencing the lawn at Rs 20 per metre.

Answer:

Let the length and breadth of the lawn be $5 x m and 3 x m$ , respectively.

Now,

Area of the lawn $= 5 x \times 3 x = 5 x^{2}$
$\Rightarrow 15 x^{2} = 3375 \Rightarrow x = \sqrt{\frac{3375}{15}} \Rightarrow x = \sqrt{225} = 15$

$Length = 5 x = 5 \times 15 = 75 m Breadth = 3 x = 3 \times 15 = 45 m$

∴ Perimeter of the lawn $= 2 (Length + Breadth) = 2 (75 + 45) = 2 \times 120 = 240 m$
Total cost of fencing the lawn at Rs 20 per metre $= 240 \times 20 = Rs 4800$

Page No 791:

Question 12:

Find the area of a rhombus each side of which measures 20 cm and one of whose diagonals is 24 cm.

Answer:

Given:
Sides are 20 cm each and one diagonal is of 24 cm.
The diagonal divides the rhombus into two congruent triangles, as shown in the figure below.

We will now use Hero's formula to find the area of triangle ABC.
First, we will find the semiperimeter.
$s = \frac{1}{2} (a + b + c) = \frac{1}{2} (20 + 20 + 24) = \frac{64}{2} = 32 m$

$Area of ∆ ABC = \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{32 (32 - 20) (32 - 20) (32 - 24)} = \sqrt{32 \times 12 \times 12 \times 8} = \sqrt{36864} = 192 {cm}^{2}$

Now,
Area of the rhombus $= 2 \times Area of triangle ABC = 192 \times 2 = 384 {cm}^{2}$

Page No 791:

Question 13:

Find the area of a trapezium whose parallel sides are 11 cm and 25 cm long and non-parallel sides are 15 cm and 13 cm.

Answer:

We will divide the trapezium into a triangle and a parallelogram.

Difference in the lengths of parallel sides $= 25 - 11 = 14 cm$
We can represent this in the following figure:

Trapezium ABCD is divided into parallelogram AECD and triangle CEB.

Consider triangle CEB.

In triangle CEB, we have:

E B = 25 - 11 = 14 cm

Using Hero's theorem, we will first evaluate the semiperimeter of triangle CEB and then evaluate its area.

Semiperimeter,

s = \frac{1}{2} (a + b + c) = \frac{1}{2} (15 + 13 + 14) = \frac{42}{2} = 21 cm

Area of triangle C E B = \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{21 (21 - 15) (21 - 13) (21 - 14)} = \sqrt{21 \times 6 \times 8 \times 7} = \sqrt{7056} = 84 {cm}^{2}

Also,

Area of triangle CEB

= \frac{1}{2} (Base \times Height)

Height of triangle CEB

= \frac{Area \times 2}{Base} = \frac{84 \times 2}{14} = 12 cm

Consider parallelogram AECD.

Area of parallelogram AECD

= Height \times Base = A E \times C F = 12 \times 11 = 132 {cm}^{2}

Area of trapezium ABCD

= Ar (∆ B E C) + Ar (parallelogram A E C D) = 132 + 84 = 216 {cm}^{2}

Page No 791:

Question 14:

The adjacent sides of a ∥gm ABCD measure 34 cm and 20 cm and the diagonal AC is 42 cm long. Find the area of the ∥gm.

Answer:

The diagonal of a parallelogram divides it into two congruent triangles. Also, the area of the parallelogram is the sum of the areas of the triangles.

We will now use Hero's formula to calculate the area of triangle ABC.

$Semiperimeter, s = \frac{1}{2} (34 + 20 + 42) = \frac{1}{2} (96) = 48 cm$

$Area of ∆ ABC = \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{48 (48 - 42) (48 - 34) (48 - 20)} = \sqrt{48 \times 6 \times 14 \times 28} = \sqrt{112896} = 336 {cm}^{2}$

Area of the parallelogram $= 2 \times Area ∆ ABC = 2 \times 336 = 672 {cm}^{2}$

Page No 791:

Question 15:

The cost of fencing a square lawn at Rs 14 per metre is Rs 2800. Find the cost of mowing the lawn at Rs 54 per 100 cm².

Answer:

Given:
Cost of fencing = Rs 2800
Rate of fencing = Rs 14

Now,
Perimeter $= \frac{Total cost}{Rate} = \frac{2800}{14} = 200 m$

Because the lawn is square, its perimeter $is 4 a, where a is the side of the square)$ .
$\Rightarrow 4 a = 200 \Rightarrow a = \frac{200}{4} = 50 m$

Area of the lawn = ${Side}^{2} = 50^{2} = 2500 m^{2}$

Cost for mowing the lawn per 100 m²= Rs 54

Cost for mowing the lawn per 1 m² $= Rs \frac{54}{100}$

Total cost for mowing the lawn per 2500 m² $= \frac{54}{100} \times 2500 = Rs 1350$

Page No 791:

Question 16:

Find the area of quad. ABCD in which AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm and diag. BD = 20 cm.

Answer:

Quadrilateral ABCD is divided into triangles $∆$ ABD and $∆$ BCD.
We will now use Hero's formula.
For $∆$ ABD:
$Semiperimeter, s = \frac{1}{2} (42 + 20 + 34) = \frac{96}{2} = 48 cm$

$Area of ∆ ABD = \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{48 (48 - 42) (48 - 34) (48 - 20)} = \sqrt{48 \times 6 \times 14 \times 28} = \sqrt{112896} = 336 {cm}^{2}$

For $∆$ BCD:
$s = \frac{1}{2} (20 + 21 + 29) = \frac{70}{2} = 35 cm$

$Area of ∆ BCD = \sqrt{s (s - a) (s - b) (s - c)} = \sqrt{35 (35 - 20) (35 - 21) (35 - 29)} = \sqrt{35 \times 15 \times 14 \times 6} = \sqrt{44100} = 210 {cm}^{2}$

Thus, we have:
Area of quadrilateral ABCD $= Ar (∆ A B D) + Ar (B D C) = 336 + 210 = 546 {cm}^{2}$

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Question 17:

A parallelogram and a rhombus are equal in area. The diagonals of the rhombus measure 120 m add 44 m. If one of the sides of the ∥gm is 66 m long, find its corresponding altitude.

Answer:

Area of the rhombus $= \frac{1}{2} (Product of diagonals) = \frac{1}{2} (120 \times 44) = 2640 m^{2}$

Area of the parallelogram $= Base \times Height = 66 \times Height$

Given:
The area of the rhombus is equal to the area of the parallelogram.

$Thus, we have : 66 \times Height = 2640 \Rightarrow Height = \frac{2640}{66} = 40 m$

∴ Corresponding height of the parallelogram = 40 m

Page No 791:

Question 18:

The diagonals of a rhombus are 48 cm and 20 cm long. Find the perimeter of the rhombus.

Answer:

Diagonals of a rhombus perpendicularly bisect each other. The statement can help us find a side of the rhombus. Consider the following figure.

ABCD is the rhombus and AC and BD are the diagonals. The diagonals intersect at point O.

We know:
$∠ D O C = 90 °$
$D O = O B = \frac{1}{2} D B = \frac{1}{2} \times 48 = 24 cm$
Similarly,
$A O = O C = \frac{1}{2} A C = \frac{1}{2} \times 20 = 10 cm$

Using Pythagoras' theorem in the right-angled triangle $∆$ DOC, we get:

$D C^{2} = \sqrt{D O^{2} + O C^{2}} = \sqrt{24^{2} + 10^{2}} = \sqrt{576 + 100} = \sqrt{676} = 26 cm$

DC is a side of the rhombus.
We know that in a rhombus, all sides are equal.
∴ Perimeter of ABCD $= 26 \times 4 = 104 cm$

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Question 19:

The adjacent sides of a parallelogram are 36 cm and 27 cm in length. If the distance between the shorter sides is 12 cm, find the distance between the longer sides.

Answer:

$Area of a parallelogram = Base \times Height ∴ A B \times D E = B C \times D F \Rightarrow D E = \frac{B C \times D F}{A B} = \frac{27 \times 12}{36} = 9 cm$

∴ Distance between the longer sides = 9 cm

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Question 20:

In a four-sides field, the length of the longer diagonal is 128 m. The lengths of perpendiculars from the opposite vertices upon this diagonal are 22.7 m and 17.3 m. Find the area of the field.

Answer:

The field, which is represented as ABCD, is given below.

The area of the field is the sum of the areas of triangles ABC and ADC.

Area of the triangle ABC $= \frac{1}{2} (AC \times BF) = \frac{1}{2} (128 \times 22.7) = 1452.8 m^{2}$

Area of the triangle ADC $= \frac{1}{2} (AC \times DE) = \frac{1}{2} (128 \times 17.3) = 1107.2 m^{2}$

Area of the field = Sum of the areas of both the triangles $= 1452.8 + 1107.2 = 2560 m^{2}$

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