A traffic signal board, indicating ‘SCHOOL AHEAD’

Question. 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:

Side of traffic signal board = a

Perimeter of traffic signal board $=3 \times a$

$2 s=3 a \Rightarrow s=\frac{3}{2} a$

By Heron’s formula,

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

Area of given triangle $=\sqrt{\frac{3}{2} a\left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)}$

$=\sqrt{\left(\frac{3}{2} a\right)\left(\frac{a}{2}\right)\left(\frac{a}{2}\right)\left(\frac{a}{2}\right)}$

$=\frac{\sqrt{3}}{4} a^{2}$...(1)

Perimeter of traffic signal board = 180 cm

Side of traffic signal board $(a)=\left(\frac{180}{3}\right) \mathrm{cm}=60 \mathrm{~cm}$

Using equation (1), area of traffic signal board $=\frac{\sqrt{3}}{4}(60 \mathrm{~cm})^{2}$

$=\left(\frac{3600}{4} \sqrt{3}\right) \mathrm{cm}^{2}=900 \sqrt{3} \mathrm{~cm}^{2}$

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