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NCERT Solutions for class 9 Maths chapter 12 – Heron’s Formula


Back Exercise

Exercise 12.1

Question 1.
A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side a. Find the area of the signal board, using Heron’s formula.If its perimeter is 180 cm, what will be the area of the signal board?
Solution:
We know that, an equilateral triangle has equal sides. So, all sides are equal to a.
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.1 1

Question 2.
The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see figure). The advertisements yield an earning of ₹5000 per m2 per year. A company hired one of its walls for 3 months. How much rent did it pay?
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.1 2
Solution:
Leta = 122m,
b = 22m
c = 120m
We have, b2 + c2 = (22)2 + (120)2 = 484 + 14400 = 14884= (122)2 = a2
Hence, the side walls are in right triangled shape.
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.1 2a
byjus class 9 maths Chapter 7 Heron's Formula 7.1 2b

Question 3.
There is a slide in a park. One of its side Company hired one of its walls for 3 months.walls has been painted in some colour with a message “KEEP THE PARK GREEN AND CLEAN” (see figure). If the sides of the wall are 15 m, 11 m and 6m, find the area painted in colour.
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.1 3
Solution:
The given figure formed a triangle whose sides are
a = 15m
b = 11m,
c = 6m
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.1 3a

Question 4.
Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Solution:
Solution Let the sides of a triangle, a = 18cm b = 10 cm and c
We have, perimeter = 42 cm
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.1 4

Question 5.
Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.
Solution:
Suppose that the sides in cm, are 12x, 17x and 25x.
Then, we know that 12x + 17x + 25x = 540 (Perimeter of triangle)
⇒ 54x = 540
⇒ x=10
So, the sides of the triangle are 12 x 10 cm, 17 x 10 cm 25 x 10 cm
i.e., 120 cm, 170 cm, 250 cm
byjus class 9 maths Chapter 7 Heron's Formula 7.1 5

Question 6.
An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.
Solution:
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.1 6
Let in isosceles ∆ ABC,
AB = AC = 12 cm (Given)
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.1 6a

Exercise 12.2

Question 1.
A park, in the shape of a quadrilateral ABCD, has ∠C = 90°, AB = 9m,BC = 12m,CD = 5m and AD = 8 m.
How much area does it occupy?
Solution:
In right ∆ BCD

NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 1
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 1a

Question 2.
Find the area of a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm.
Solution:
Area of quadrilateral ABCD = Area of ∆ABC+ Area of ∆ACD
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 2
In ∆ ABC,
We have, AB = 3 cm, BC = 4 cm, CA = 5 cm
Therefore, AB2 + BC2 = 32 + 42 = 9 + 16 = 25 = (5)2 = CA2
Hence, ∆ ABC is a right triangle
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 2a

Question 3.
Radha made a picture of an aeroplane with coloured paper as shown in figure. Find the total area of the paper used.
tiwari academy class 9 maths Chapter 7 Heron's Formula 7.2 3
Solution:
For part I
It is a triangle with sides 5 cm, 5cm and 1cm
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 3a
For part II
It is a rectangle with sides 6.5 cm and 1 cm
∴ Area of part II = 6.5 x 1
(∵ Area of rectangle = Length x Breadth)
= 6.5 cm2

For part III
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 3B
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 3C

Question 4.
A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26 cm, 28 cm and 30 cm, and the parallelogram stands on the base 28 cm, find the height of the parallelogram
Solution:
Let ABC be a triangle with sides
AB = 26 cm, BC = 28 cm, CA = 30 cm
tiwari academy class 9 maths Chapter 7 Heron's Formula 7.2 4
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 4A
We know that,
Area of parallelogram = Base x Height …(i)
We have,
Area of parallelogram = Area of ∆ ABC (Given)
= 336 cm2
From Eq. (i), we have
Base x Height = 336
⇒ 28 x Height = 336
⇒ Height = 33628
⇒ Height =12cm

Question 5.
A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, how much area of grass field will each cow be getting?
Solution:
Let ABCD be a rhombus.
Area of the rhombus ABCD = 2x area of ∆ ABD …(i)
(Since, in a rhombus diagonals divides two equal parts)
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 5
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 5a

Question 6.
An umbrella is made by stitching 10 triangular pieces of cloth of two different colours (see figure), each piece measuring 20 cm, 50 cm and 50 cm. How much cloth of each colour is required for the umbrella?
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 6
Solution:
In an umbrella, each triangular piece is an isosceles triangle with sides 50 cm, 50 cm, 20 cm.
tiwari academy class 9 maths Chapter 7 Heron's Formula 7.2 6a
Since, there are 10 triangular piece, in those of them 5.5 are of different colours.
Hence, total area of cloth of each colour = 5 x 200√6 cm2 = 1000√6 cm2

Question 7.
A kite in the shape of a square with a diagonal 32 cm and an isosceles triangle of base 8 cm and sides 6 cm each is to be made of three different shades as shown in figure. How much paper of each shade has been used in it?
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 7
Solution:
Since, the kite is in the shape of a square
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 7a
Each diagonal of square =32 cm (Given)
We know that, the diagonals of a square bisect each other at right angle.
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 7b
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 7c
Hence, paper of I colour has been used = 256 cm2
Paper of II colour has been used = 256 cm2
Paper of III colour has been used = 17.92 cm2

Question 8.
A floral design on a floor is made up of 16 tiles which are triangular, the sides of the triangle being 9 cm, 28 cm and 35 cm (see figure). Find the cost of polishing the tiles at the rate of 50 paise per cm .
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 8
Solution:
Given, the sides of a triangular tiles are 9 cm, 28 cm and 35 cm.
For each triangular tile, we have
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 8a

Question 9.
A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non-parallel sides are 14 m and 13 m. Find the area of the field.
Solution:
Here, ABCD is a trapezium and AB || DC.
tiwari academy class 9 maths Chapter 7 Heron's Formula 7.2 9
NCERT Solutions for Class 9 Maths Chapter 7 Heron's Formula 7.2 9a