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 1

✯✯ 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?

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✰✰ ANSWER ✰✰

$\longmapsto{Perimeter=180cm}$

$\longmapsto{Let\:three\:sides=a,a,a}$

➥A.T.Q : -

$\longmapsto{a+a+a=180}$

$\longmapsto{3a=180}$

$\longmapsto{a=\cancel\dfrac{180}{3}}$

$\pink\longmapsto\:\large\underline{\boxed{\bf\red{a}\blue{=}\green{60cm}}}$

➥Now ,

$\longmapsto{S=\dfrac{a+b+c}{2}}$

$\longmapsto{\dfrac{60+60+60}{2}}$

$\longmapsto{\cancel\dfrac{180}{2}}$

$\red\longmapsto\:\large\underline{\boxed{\bf\green{s}\orange{=}\purple{90cm}}}$

➥So ,

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

$\longmapsto{\sqrt{90(90-60)(90-60)(90-60)}}$

$\longmapsto{\sqrt{90\times{30}\times{30}\times{30}}}$

$\longmapsto{\sqrt{3\times{3}\times{2}\times{5}\times{2}\times{3}\times{5}\times{2}\times{3}\times{5}\times{2}\times{3}\times{5}}}$

$\longmapsto{3\times{2}\times{5}\times{3}\times{2}\times{5}\sqrt{3}}$

$\longmapsto{\large{\boxed{\bold{\bold{\purple{\sf{900\sqrt{3}cm.}}}}}}}$

➥So , The Area of Signal Board is 900√3 cm...

Answer 2

$\huge\star{\red{\underline{Answer:}}}$

$\bf{\purple{\underline{Given:}}}$

Perimeter of triangle = 180 cm

$\bf{\pink{\underline{To\;Find:}}}$

Area of the triangle

$\bf{\blue{\underline{Solution:}}}$

sides of triangle are a,a and a units

So, $\sf{s=\frac{a+a+a}{2}=\frac{3a}{2}}$

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

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

Perimeter of equilateral triangle =3a

according to the question,

$\sf{3a=180cm \implies a = \frac{180}{3}= 60 cm}$

Therefore,area of equilateral triangle:

$\sf{\frac{{a}^{2}}{4}\sqrt{3} = \frac{{60}^{2}}{4}\sqrt{3}}$

$\sf{\frac{\cancel 3600}{\cancel 4}\sqrt{3} \implies 900\sqrt{3}}$

hence, the area of equilateral triangle is

$\boxed{\boxed{\boxed{=900\sqrt{3}{cm}^{2}}}}$