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?
✓
Answer
A traffic signal board is an equilateral triangle with side a.
Perimeter of the signal board,
2s = a + a + a
$\Rightarrow \mathrm{s}=\frac{3}{2} a$
Area of triangle $=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{\frac{3 a}{2}\left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)\left(\frac{3}{2} a-a\right)}$
$=\sqrt{\frac{3 a}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}}=\sqrt{\frac{3 a^{4}}{16}}=\frac{\sqrt{3}}{4} a^{2}$ sq. units
Now, if perimeter = 180 cm
3a = 180
$\Rightarrow$ a = 60 cm
$\therefore \text { Area of signal board }= \frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times(60)^{2}=900 \sqrt{3} \mathrm{cm}^{2}$
So, area of the signal board is $900 \sqrt{3} \mathrm{cm}^{2}$.
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