Question

An isosceles triangle has perimeter 30cm and each of its equal sides is 12cm . Find its area (use √15 =3.88)​

Answer 1

$\huge\frak{AnswEr :}$

Given Parameters :

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• Perimeter = 30 cm

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• Length of equal sides = 12cm

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Let us assume that the third side of isosceles triangle be x cm.

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${\underline{\sf \bigstar \: According \: to \: the \: question :}}$

$\implies\sf \:x + 12 + 12 = 30 \\ \\ \\ \implies\sf \:x + 24 = 30 \\ \\ \\ \implies\sf \:x = 30 - 24 \\ \\ \\ \implies \large{ \boxed{\frak{ \purple{ \:x = 6}}}}$

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Now,

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By using Heron's formula :

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$\implies\sf \: s = \dfrac{a + b + c}{2} \\ \\ \\ \implies\sf \: s = \dfrac{Perimeter}{2} \\ \\ \\ \implies\sf \: s = \frac{30}{2} \\ \\ \\ \implies \large{ \boxed{ \frak{\pink{\: s = 15}}}}$

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We know that,

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

Substituting the values :

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$\implies\sf \: Area \: of \: \triangle = \sqrt{15(15 - a)(15 - b)(15 - c)} \\ \\ \\ \implies\sf \: Area \: of \: \triangle =\sqrt{15(15 - 12)(15 - 12)(15 - 6)} \\ \\ \\ \implies\sf \: Area \: of \: \triangle = \sqrt{15 \times 3 \times 3 \times 9} \\ \\ \\ \implies\sf \: Area \: of \: \triangle = 3 \times 3 \times \sqrt{15} \\ \\ \\ \implies\sf \: Area \: of \: \triangle = 9 \times \sqrt{15} \\ \\ \\ \implies\sf \: Area \: of \: \triangle = 9 \times 3.88 \\ \\ \\ \implies \large{ \boxed{ \frak{\blue{ \: Area \: of \: \triangle = 34.92 \: cm {}^{2} }}}}$

$\therefore \: {\underline{\sf{ Area \: of \: the \: isosceles \: \triangle \: is \: \bf \: 34.92 cm{}^{2}. }}}$

Answer 2

$\sf{\bold{\purple{\star{\underline{\underline{Given}}}}}}$

• Perimeter = 30cm
• equal sides = 12cm

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$\sf{\bold{\purple{\star{\underline{\underline{To\: Find }}}}}}$

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• Area = ??

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$\sf{\bold{\purple{\star{\underline{\underline{Solution}}}}}}$

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• let the unknown side be x

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$\sf{\star{\boxed{\orange{\frak{ Perimeter = sum \: of \: all \: sides }}}}}$

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$\sf : \implies\:{\frak{ 30 = 12 + 12 + x }}$

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$\sf : \implies\:{\frak{ 30 = 24 + x }}$

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$\sf : \implies\:{\frak{ x = 30 - 24}}$

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$\sf : \implies\:{\frak{ x = 6 }}$

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• All the sides of the triangle are 6cm , 12cm and 12cm

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$\sf{\star{\boxed{\orange{\frak{ Semi\: perimeter = \dfrac{perimeter}{2} }}}}}$

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$\sf : \implies\:{\frak{ Semi perimeter = \dfrac{30}{2}}}$

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$\sf : \implies\:{\frak{ Semi\: perimeter = 15cm }}$

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$\sf{\star{\boxed{\orange{\frak{ Area = \sqrt{s(s-a)(s-b)(s-c)} }}}}}$

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$\sf : \implies\:{\frak{ Area = \sqrt{ 15 ( 15 - 12 ) ( 15 - 12 )( 15 - 6) }}}$

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$\sf : \implies\:{\frak{ Area = \sqrt{15\times 3\times 3 \times 9}}}$

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$\sf : \implies\:{\frak{ Area = \sqrt{15\times 3\times 3 \times 3\times 3 }}}$

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$\sf : \implies\:{\frak{ Area = 9 \sqrt{15}}}$

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$\sf : \implies\:{\frak{ Area = 9\times 3.88}}$

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$\sf : \implies\:{\frak{ Area = 34.92cm^2}}$

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$\sf{\star{\boxed{\green{\frak{ Area = 34.92cm^2 }}}}}$

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