Question

an isosceles tirangles has perimeter 60cm and each ofthe equal sides are 10cm .Find the area triangle

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Answer 1

Correct Question :

An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

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Answer :

• The area of an isosceles triangle = 9√15 cm².

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Given :

• The perimeter of an isosceles triangle = 30 cm.
• The equal sides = 12 cm.

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To Find :

• The area of an isosceles triangle.

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Solution :

First, we need to find the unequal side.

Isosceles triangle :– It is a triangle in which 2 sides area equal and 1 side is unequal.

Given that,

• The perimeter of an isosceles triangle = 30 cm.

$\implies a + b + c = 30 \: cm$

• The equal sides = 12 cm.

$\implies12 \: cm + 12 \: cm + c = 30 \: cm \\ \\ \implies24 \: cm + c = 30 \: cm \\ \\ \implies c = 30 \: cm - 24 \: cm \\ \\ \implies c =6 \: cm$

Hence, the third side of an isosceles triangle is 40 cm.

Now, we have to find the area of an isosceles triangle.

By heron's formula,

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

Where,

s = semi-perimeter

$\boxed{ \orange{ \sf{s = \frac{a + b + c}{2} }}}$

We have,

• a = 12 cm.
• b = 12 cm.
• c = 6 cm.

$\implies s = \dfrac{12 \: cm + 12 \: cm +6 \: cm }{2} \\ \\ \implies s = \dfrac{24 \: cm + 6 \: cm}{2} \\ \\ \implies s = \cancel \dfrac{30}{2} \: cm \\ \\ \implies s = \dfrac{15}{1} \: cm \\ \\ \implies s = 15 \: cm$

Hence, the semi-perimeter (s) is 30 cm.

Now we have,

• a = 12 cm.
• b = 12 cm.
• c = 6 cm.
• s = 15 cm.

Now, substitute all the values in the heron's formula.

$\implies \sqrt{s(s - a)(s - b)(s - c)} \\ \\ \implies \sqrt{15 \: cm(15 \: cm - 12 \: cm)(15 \: cm - 12 \: cm)(15 \: cm - 6 \: cm)} \\ \\ \implies \sqrt{15(3)(3)(9) \: {cm}^{4} } \\ \\ \implies \sqrt{(3 \times 5)(3)(3)(3 \times 3) \: {cm}^{4} } \\ \\ \implies \sqrt{3 \times 5 \times 3 \times3 \times 3 \times 3 } \: {cm}^{2} \\ \\ \implies \sqrt{3 \times 3 \times 3 \times 3 \times 3 \times 5} \: {cm}^{2} \\ \\ \implies3 \times 3 \sqrt{3 \times 5} \: {cm}^{2} \\ \\ \implies9 \sqrt{15} \: {cm}^{2}$

Hence,

The area of an isosceles triangle is 9√15 cm².