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[Solved] 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?

  

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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?

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Given,

Side of the signal board = a

Perimeter of the signal board = 3a

= 180 cm

∴ a = 60 cm

Semi perimeter of the signal board (s) = 3a/2

By using Heron’s formula,

Area of the triangular signal board will be

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

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

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

= \(\sqrt{\frac{3a^4}{16}}\)

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

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

= 900\(\sqrt{3}cm^2\)

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