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Chapter 12

Herons Formula

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Heron's Formula 

Heron's formula is a significant concept covered in the CBSE Class 9 Maths syllabus in the first semester. Therefore, grasping the concept is crucial. One of the best methods to do so is to use the NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula. These answers were developed by experienced teachers based on the current CBSE term-by-term syllabus. The NCERT Solutions for Class 9 include detailed and step-by-step explanations for all of the solutions provided in this chapter's exercises.

Exercise Questions

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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: 

Given that the signal board's side = a 

The signal board's perimeter is 3a = 180 cm. 

Therefore, a = 60 cm 

The signal board's semi-perimeter (s)

= 3a/2 

Using Heron's formula, you can 

The triangle signal board's area will be

= \( \sqrt{s (s-a) (s- b) (s-c)} \)

= \( \sqrt{\frac{3a}{2} (\frac{3a}{2}-a) (\frac{3a}{2}- a) (\frac{3a}{2} - a)} \)

= \( \sqrt{\frac{3a}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}} \)

=  \( \sqrt{\frac{3a^{4}}{16}} \)

=  \( \sqrt{3} \frac{a^{2}}{16} \)

=  \( \sqrt{3} \frac{a^{2}}{4} \)

=  \( \frac{\sqrt{3}}{4} \times 60 \times 60 \)

= 900\( \sqrt{3} \) cm2

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 Fig. 12.9). 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?

Solution:

The triangle ABC has sides of 122 m, 22 m, and 120 m, respectively. 

The perimeter will now be (122 + 22 + 120) 

= 264 m. 

Additionally, the semi-perimeter (s) = 264/2 

= 132 m. 

Using Heron's formula as a guide, 

The triangle area is 

= \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{132 (132-122) (132-22) (132-120)} ~m^{2} \)

= \( \sqrt{132 \times 10 \times 110 \times 12}~ m^{2} \)

= \( \sqrt{1742400}~m^{2} \)

= 1320 \( m^{2} \)

We know that the yearly advertising rent is approximately 5000 per \( m^{2} \). 

As a result, the three-month rent for one wall is Rs. (1320 \( \times \) 5000 \( \times \) 3)/12 = Rs. 16,50,000.

Question 3. There is a slide in a park. One of its side walls has been painted in some colour with a message “KEEP THE PARK GREEN AND CLEAN” (see Fig. 12.10 ). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.

Solution:

The dimensions of the wall are 15 metres, 11 metres, and 6 metres.

As a result, the semi-perimeter (s) of a triangle wall is (15+11+6)/2 m 

= 16 m. 

Using Heron's formula as a guide, 

Area of the message 

= \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{16 (16-15) (16-11) (16-6)}~ m^{2} \)

= \( \sqrt{16 \times 1 \times 5 \times 10}~m^{2} \)

= \( \sqrt{800}~m^{2} \)

= 20 \( \sqrt{2}~ m^{2} \)

Question 4. Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42cm.

Solution: Assume that the triangle's third side is "x." 

The triangle's three sides are now 18 cm, 10 cm, and "x" cm. 

The triangle's perimeter is calculated to be 42 cm. 

As a result, x = 42-18+10 cm 

= 14 cm. 

As a result, the triangle's semi-perimeter = 42/2

 = 21 cm. 

Using Heron's formula as a guide,  

The triangle's area,

= \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{21 (21-18) (21-10) (21-14)}~cm^{2} \)

= \( \sqrt{21 \times 3 \times 11 \times 7}~ cm^{2} \)

= 21 \( \sqrt{11}~ cm^{2} \)

Question 5. Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540cm. Find its area.

Solution:

The sides of the triangle have a ratio of 12: 17: 25. 

Let the common ratio between the triangle's sides be "x." 

As a result, the sides are 12x, 17x, and 25x, respectively. 

It is also assumed that the triangle's circumference is 540 cm. 

12x+17x+25x = 540 cm

54 \( \times \) 54 cm = 540 cm

As a result of this, 

x = 10.

The triangle's sides are now 120 cm, 170 cm, and 250 cm. 

As a result, the triangle's semi-perimeter (s) = 540/2 

= 270 cm. 

Using Heron's formula as a guide, 

Triangle area,

= \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{270 (270-170) (270-150) (270-250)}~ cm^{2} \)

= \( \sqrt{270 \times 150 \times 100 \times 20}~ cm^{2} \)

= 9000 \( cm^{2} \)

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:

Allow the third side to be x at first. 

It is assumed that the equal sides' length is 12 cm and the perimeter is 30 cm. 

Thus, 30 = 12 + 12 + x 

As a result, the third side's length equals 6 cm. 

As a result, the isosceles triangle's semiperimeter (s) = 30/2 cm 

= 15 cm. 

Using Heron's formula as a guide, 

Triangle area,

= \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{15 (15-12) (15-12) (15-6)} cm^{2} \)

= \( \sqrt{15 \times 3 \times 3 \times 9} ~cm^{2} \)

= \( \sqrt{1215}~ cm^{2} \)

= \( 9 \sqrt{15}~ cm^{2} \)

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

Solution:

To start, make a quadrilateral ABCD and join BD. 

We know 

C = 90 °, AB = 9 metres, BC = 12 metres, CD = 5 metres, and AD = 8 metres 

The diagram is as follows:

Apply Pythagoras theorem to ΔBCD now. 

\( BC^{2} \)+ \( CD^{2} \) = \( BD^{2} \)

\( BD^{2} \) = 122+52 

\( BD^{2} \) = 169 

BD = 13 m 

Now, the area of ΔBCD is equal to (½ \( \times \)12 \( \times \) 5) = 30 \( m^{2} \). 

ΔABD's semi-perimeter 

(perimeter/2) = (s) 

= (8 + 9 + 13)/2 m 

= 30/2 m 

= 15 m 

Using Heron's formula as a guide, 

Area of ΔABD

= \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{15 (15-13) (15-9) (15-8)}~ m^{2} \)

= \( \sqrt{15 \times 2 \times 6 \times 7} \)

= \( \sqrt{1260} \)

= 6\( \sqrt{35} \) \( m^{2} \)

= 35.5 \( m^{2} \) (approximately) 

\( \therefore \) ABCD quadrilateral area = ΔBCD quadrilateral area + Area of ΔABD 

= 30 \( m^{2} \) + 35.5 \( m^{2} \)

= 65.5 \( m^{2} \)

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:

Make a diagram with the parameters you've been given. 

Now, in ΔABC, apply the Pythagorean theorem. 

\( AC^{2} = AB^{2} + BC^{2} \)

32+42 = 52

 = 25

As a result, we can deduce that ΔABC is right-angled at B. 

As a result, BCD has a surface area of (1/2 \( \times \) 3 \( \times \) 4) = 6 \( cm^{2} \). 

Semi-perimeter ACD (s) = (perimeter/2) 

= (5+5+4)/2 cm 

= 14/2 cm 

= 7 m

Using Heron's formula as a guide, 

Area of ΔACD 

= \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{7 (7-5) (7-5) (7-4)} ~cm^{2} \)

= \( \sqrt{7 \times 2 \times 2 \times 3}~ cm^{2} \)

= \( \sqrt{ 84 }~cm^{2} \)

= 9.17 \( cm^{2} \)

= 221 \( cm^{2} \)  (approximately) 

ABCD quadrilateral area = ΔABC area + Δ ACD area = 6 \( cm^{2} \)  + 9.17 \( cm^{2} \)

= 15.17 \( cm^{2} \)

Question 3. Radha made a picture of an aeroplane with coloured paper as shown in Fig 12.15. Find the total area of the paper used.

Solution: 

For Triangle I Section,

The sides of the isosceles triangle are 5 cm, 1 cm, and 5 cm. 

Perimeter = 5+5+1 = 11 cm 

As a result, the semi-perimeter = 1 1/2 cm = 5.5 cm. 

Using Heron's formula as a guide, 

Area =  \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{5.5. (5.5-5) (5.5-5) (5.5-1)} cm^{2} \)

= \( \sqrt{5.5 \times 0.5 \times 0.5 \times 4.5}~ cm^{2} \)

= 6.1875 \( cm^{2} \)

= 0.75 \( \times \) 3.317 \( cm^{2} \) 

= 2.488 \( cm^{2} \) (approx) 

For the quadrilateral II section, 

use the following formula: 

The length and breadth of this quadrilateral are 6.5 cm and 1 cm, respectively. 

\( \therefore \) Area = 6.5\( \times \)1 \( cm^{2} \) = 6.5 \( cm^{2} \)

For the quadrilateral III section, 

use the following formula: 

It's a trapezoid with two 1 cm sides and a 2 cm third side.

Area of the trapezoid = Area of the parallelogram + Area of the equilateral triangle

The parallelogram's perpendicular height will be 

= \( \sqrt{1^{2} - 0.5^{2}} \)

= 0.86 cm. 

And the equilateral triangle's area will be (3/4) a2 = 0.43. 

The trapezoid's area is 0.86 + 0.43 = 1.3 cm2 (approximately). 

For triangles IV and V, use the following formula: 

These are two congruent right-angled triangles with a base of 6 cm and a height of 1.5 cm. 

Area of triangles IV and V = 2(1/2 x 6 x 1.5) \( cm^{2} \) = 9 \( cm^{2} \)

As a result, the total area of the paper used is 19.3 \( cm^{2} \) (2.488 + 6.5 + 1.3 + 9) \( cm^{2} \).

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:

Now, 

It is assumed that the areas of the parallelogram and triangle are equal. 

The triangle's sides are 26 cm, 28 cm, and 30 cm long. 

As a result, the perimeter equals 26+28+30 = 84 cm. 

Its semi-perimeter is 84/2 cm, which equals 42 cm. 

Now, using Heron's formula, the triangle's area =

= \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{42 (42-26) (42-28) (42-30)}~ cm^{2} \)

=  \( \sqrt{42 \times 16 \times 14 \times 12}~cm^{2} \)

= \( \sqrt{112896}~cm^{2} \)

= 336 \( cm^{2} \)

Let h be the height of the parallelogram. 

Because the area of a parallelogram equals the area of a triangle, 

= 336 \( cm^{2} \times \) 28 cm h 

\( \therefore \) h = 336/28 cm 

Thus, the parallelogram's height is 12 cm.

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: First, make a rhombus-shaped field with ABCD vertices. The rhombus is divided into two congruent triangles with equal areas by the diagonal AC. The following is a diagram.

Consider the BCD triangle. 

(48 + 30 + 30)/2 m = 54 m is its semi-perimeter. 

Using Heron's formula as a guide, 

Area of the ΔBCD = \( \sqrt{s (s-a) (s-b) (s-c)} \)

= \( \sqrt{54 (54-48) (54-48) (54-30)} \)

= \( \sqrt{54 \times 6 \times 24 \times 24} \)

= \( \sqrt{186624} \)

= 432 \( m^{2} \)

 Therefore, Field area = 2 \( \times \) ΔBCD area = \( (2 \times 432) m^{2} \) = 864 \( m^{2} \)

As a result, each cow will receive an area of the grass field equal to (864/18) \( m^{2} \) = 48 \( m^{2} \).

These were NCERT solutions for Class 9 Maths Chapter 12 Heron’s Formula. We hope you have understood all the problems and their solutions. If you need NCERT solutions of other maths and science topics, you may find them on the MSVGo app. 

Our specialist faculty team has articulated the solutions to all NCERT problems in order to help students develop their problem-solving abilities. You can also refer to the study materials provided on MSVGo to gain a better understanding of Heron's Formula and other such complex topics.

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