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 it's perimeter is 180 cm. what will be the area of the signal board?

Answer 1

2s perimeter of triangle = 108cm
side of triangle. = a
semi - perimeter = s = a+b+c/2
s = a+a+a/2
s = 3a / (2s perimeter =108) 2s = 3a
108 = 3a
a = 108/ 3
a = 60
according to heron's formula
area of triangle = √s(s-a)(s-b)(s-c)
= √3a/2 ( 3a/2 - a/1)(3a/2 - a/1)(3a/2-a/1)
=√3a/2(a/2)(a/2)(a/2)
= √3a^4/16
= a^2/4√3
= a^2×√3
= 60×60/4×√3 (a= 60cm)
= 900√3 m^2

Answer 2

☯ Let 2s be the perimeter of the signal board. Then,

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$:\implies\sf 2s = a + a + a$

$:\implies\sf 2s = 3a$

$:\implies\sf s = \dfrac{3a}{2}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

☯ Let A be the area of the given equilateral triangle, Then,

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Using Heron's Formula,

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$\star\;{\boxed{\sf{\purple{A = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

$:\implies\sf A = \sqrt{ \dfrac{3a}{2} \bigg( \dfrac{3a}{2} - a \bigg) \bigg( \dfrac{3a}{2} - a \bigg) \bigg( \dfrac{3a}{2} - a \bigg)}\qquad\qquad\bigg\lgroup \because\;a = b = c\bigg\rgroup$

$:\implies\sf A = \sqrt{ \dfrac{3a}{2} \times \dfrac{a}{2} \times \dfrac{a}{2} \times \dfrac{a}{2}}$

$:\implies\sf A = \sqrt{ \dfrac{3a^4}{16}}$

$:\implies{\boxed{\frak{\pink{A = \dfrac{ \sqrt{3}}{4}a^2}}}}\;\bigstar$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

If, Perimeter = 180 cm. Then,

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$:\implies\sf 2s = 180$

$:\implies\sf \cancel{2} \times \dfrac{3a}{ \cancel{2}} = 180$

$:\implies\sf 3a = 180$

$:\implies{\boxed{\frak{\purple{a = 60}}}}\;\bigstar$

Therefore,

⠀⠀

$:\implies\sf A = \dfrac{ \sqrt{3}}{4} \times (60)^2$

$:\implies\sf A = \dfrac{ \sqrt{3}}{ \cancel{4}} \times \cancel{3600}$

$:\implies{\boxed{\frak{\pink{A = 900 \sqrt{3}\;cm^2}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;Area\;of\;signal\;board\;is\: \bf{900 \sqrt{3}\;cm^2}.}}}$