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: Dimensions of the traffic signal board (equilateral triangle) and its perimeter.
By using Heron’s formula, we can calculate the area of a triangle.
Heron's formula for the area of a triangle is: √[s(s - a)(s - b)(s - c)]
Where a, b and c are the sides of the triangle and s = Semi-perimeter = Half the Perimeter of the triangle
Each side of traffic signal board (equilateral triangle) = ‘a’ cm
Perimeter of traffic signal board (equilateral triangle) = sum of all the sides = a + a + a = 3a
Semi Perimeter, s = (a + b + c)/2 = (a + a + a)/2 = 3a/2
By using Heron’s formula,
Area of a triangle = √[s(s - a)(s - b)(s - c)]
Area of a (triangle) traffic signal board
= √[s(s - a)(s - b)(s - c)]
= √[s(s - a)(s - a)(s - a)] {since all three sides are equal to "a"}
= (s - a) √[s(s - a)].....(1)
We know, s = 3a/2, so substituting this value in equation (1)
Area = (3a/2 - a)√[3a/2(3a/2 - a)]
= (a/2) √[3a/2(a/2)]
= a/2 × a/2 × √3
= (√3/4)a2
Area of the signal board = (√3/4)a2 sq. units
Now given perimeter = 180 cm
Each side of triangle = 180/3 cm
a = 60 cm
Substituting the value of 'a'.
Area of the signal board = (√3/4)(60)2
= (√3/4)(3600)
= 900√3
Area of the signal board = 900√3 cm2
☛ Check: NCERT Solutions for Class 9 Maths Chapter 12
Video Solution:
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?
Class 9 Maths NCERT Solutions Chapter 12 Exercise 12.1 Question 1
Summary:
A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. The area of the signal board, using Heron’s formula if its perimeter is 180 cm is 900√3 cm2.
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