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Question 1:

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

Solution 1:

Ratio of the sides of the triangle = 12 : 17 : 25

Let the common ratio be x then sides are 12x, 17x and 25x

Perimeter of the triangle = 540cm

12x + 17x + 25x = 540 cm

⇒ 54x = 540cm

⇒ x = 10

Sides of triangle are,

12x = 12 × 10 = 120cm

17x = 17 × 10 = 170cm

25x = 25 × 10 = 250cm

Semi perimeter of triangle(s) = 540/2 = 270cm

Using Heron's formula,

Comments

Reetika

Question 2:

An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

Solution 2:

Length of the equal sides = 12cm

Perimeter of the triangle = 30cm

Length of the third side = 30 - (12+12) cm = 6cm

Semi perimeter of the triangle(s) = 30/2 cm = 15cm

Using heron's formula,

Area of the triangle = √s (s-a) (s-b) (s-c)

= √15(15 - 12) (15 - 12) (15 - 6) $cm^{2}$

= √15 × 3 × 3 × 9 $cm^{2}$

= 9√15 $cm^{2}$

Comments

Reetika

Question 3:

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 3:

The perimeter of a triangle is equal to the sum
of its three sides it is denoted by 2S.
2s=(a+b+c)
s=(a+b+c)/2
Here ,s is called semi perimeter of a triangle.

The formula given by Heron about the area of a
triangle is known as Heron's formula. According to this formula area of a
triangle= √s (s-a) (s-b) (s-c)
Where a, b and c are three sides of a triangle
and s is a semi perimeter.

This formula can be used for any triangle to
calculate its area and it is very useful when it is not possible to find the height
of the triangle easily . Heron's formula is
generally used for calculating area of scalene triangle.
____________________________________________________________
Solution:Given, side of a signal whose shape is an equilateral triangle= a

Semi perimeter, s=a+a+a/2= 3a/2

Using heron’s formula,
Area of the signal board = √s (s-a) (s-b) (s-c)= √(3a/2) (3a/2 – a) (3a/2 – a) (3a/2 – a)
= √3a/2 × a/2 × a/2 × a/2
= √3aâ?´/16
= √3a²/4

Hence, area of signal board with side a by using Herons formula is= √3a²/4

Now,Perimeter of an equilateral triangle= 3a
Perimeter of the traffic signal board = 180 cm

(given) 3a = 180 cm a = 180/3= 60 cm

Now , area of signal board=√3a²/4
= √3/4 × 60 × 60 = 900√3 cm²

Hence , the area of the signal board when perimeter is 180 cm is 900√3 cm².

Comments

Reetika

Question 4:

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 Rs.5000 per $m^{2}$ per year. A company hired one of its walls for 3 months. How much rent did it pay?

Solution 4:

Let the sides of the triangle are

a=122 m, b=22 m & c= 120 m.
Semi Perimeter of the triangle

s = (a+b+c) /2
s=(122 + 22 + 120) / 2
s= 264/2= 132m

By Heron’s formula,

Given, earning on 1m² per year= Rs. 5000
Earning on 1320 m² per year=1320×5000= Rs. 66,00,000/-
Now, earning in 1320 m² in 12 months= Rs. 66,00,000/-

earning in 3 months = $Rs. 6600000\times \frac{3}{12}= 1650000$
Hence, the rent paid by the company for 3 months is Rs. 16,50,000/-.

Comments

Reetika

Question 5:

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 Figure ). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.

Solution 5:

Here area painted in colour will be equal to the area of triangle with side 15m,11m and 6 m
Let the Sides of the triangular wall are a=15 m, b=11 m & c=6 m.
Semi Perimeter of the â??,s = (a+b+c) /2
Semi perimeter of triangular wall (s) = (15 + 11 + 6)/2 m = 16 m
Using heron’s formula,
Area of the wall = √s (s-a) (s-b) (s-c)
= √16(16 – 15) (16 – 11) (16 – 6)
= √16 × 1 × 5 × 10
= √ 4×4×5×5×2
= 4×5√2
= 20√2 m²
Hence, the area painted in colour is 20√2 m²

Comments

Reetika

Question 6:

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

Solution 6:

Given,

Side a=18cm, Side b=10cm, Perimeter=42cm=a+b+c

:. Putting value 42=18+10+c

42=28+c

42-28=c

14=c

Now,S=(a+b+c)/2

:. Putting value S=42/2S=21

Now according to Heron's formula

Area of a triangle=√{s(s-a)(s-b)(s-c)}

:. Putting value=√{21(21-18)(21-10)(21-14)}

=√{21(3)(11)(7)}

=√4851=21√11cm²

Comments

Reetika

All Exercises - Chapter 12 - Herons Formula

Exercise 12.1

Exercise 12.2

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Learn Exercise 12.1 with Free Lessons & Tips

All Exercises - Chapter 12 - Herons Formula

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