Question

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a traffic signal board indicating School ahead is an equilateral triangle with side a find the area of signal board using heron's formula with the perimeter is 162 centimetres what will be the area of the signal board?​​

Answer 1

a traffic signal board indicating School ahead is an equilateral triangle with side a find the area of signal board using heron's formula with the perimeter is 162 centimetres what will be the area of the signal board?

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Answer 2

Given :-

• Side of an equilateral triangle is a
• Perimeter is 162 cm

To Find :-

• The area of the board using Heron's Formula.

Solution :-

$\small\underline{\pmb{\sf Heron's\: Formula \: is \: given \: by :-}}$

$\underline{\boxed{\sf\sqrt{s(s-a)(s-b)(s-c)}}}$

• We are given, perimeter of the triangle is 162 cm.Let's find it’s semi-perimeter.

$\sf :\implies Semi-\ Perimeter = \dfrac{162}{2}$

$\sf \green{:\implies Semi- \ Perimeter = 81}$

$\small\underline{\pmb{\sf According \: to \: the \: question :-}}$

$\sf:\implies a+a+a=162$

$\sf :\implies 3a=162$

$\sf :\implies a=\dfrac{162}{3}$

$\sf \pink{ :\implies a=54}$

$\small\underline{\pmb{\sf Using \: Heron's\: Formula :-}}$

$\begin{gathered}\;{\boxed{\sf{\purple{Area_{Signals\:board} = \sqrt{s(s - a)(s - b)(s - c)}}}}}\\ \\\end{gathered}$

$\sf :\implies Area = \sqrt{s(s-a)(s-b)(s-c)}$

$\sf :\implies Area =\sqrt{81(81-54)(81-54)(81-54)}$

$\sf :\implies Area= \sqrt{81(27)(27)(27)}$

$\sf :\implies Area =\sqrt{4(27) \times 3}$

$\sf :\implies Area= 27\times 27√3$

$\sf \sf{:\implies Area = 729\sqrt{3} }$

$\sf \sf{:\implies Area = 729\times 1.73 }$

$\sf \pink{\sf{:\implies Area = 1261.17 }}$

$\therefore\:\underline{\textsf{Area of signal board is \textbf{1261.17cm² }}}.$