The perimeter of an isosceles triangle is 32 cm.
Question:

The perimeter of an isosceles triangle is 32 cm. The ratio of the equal side to its base is 3 : 2. Find  the area of the triangle.

Solution:

The ratio of the equal side to its base is 3 : 2.
Ratio of sides = 3 : 3 : 2.
Let the three sides of triangle be 3x, 3x, 2x.
The perimeter of isosceles triangle = 32 cm.

$\Rightarrow 3 x+3 x+2 x=32 \mathrm{~cm}$

$\Rightarrow 8 x=32$

$\Rightarrow x=4 \mathrm{~cm}$

Therefore, the three side of triangle are 3x, 3x, 2x = 12 cm, 12 cm, 8 cm.

Let $S$ be the semi-perimeter of the triangle. Then, $S=\frac{1}{2}(12+12+8)=\frac{32}{2}=16$

Area of the triangle will be

$=\sqrt{S(S-a)(S-b)(S-c)}$

$=\sqrt{16(16-12)(16-12)(16-8)}$

$=\sqrt{16 \times 4 \times 4 \times 8}$

$=4 \times 4 \sqrt{8}=4 \times 4 \times 2 \sqrt{2}=32 \sqrt{2} \mathrm{~cm}^{2}$

Disclaimer: The answer does not match with the answer given in the book.

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