Question : The ratio of three sides of a triangle is $5: 5: 8$. If the area of triangle is $12\;\mathrm{cm^2}$, then what is the length (in$\;\mathrm{cm}$) of the equal sides?
Option 1: 5
Option 2: 8
Option 3: 6
Option 4: 2.5
Correct Answer: 5
Solution :
Given that the ratio of the sides of the triangle is $5:5:8$.
This is an isosceles triangle.
Let the sides of the triangle be $5x$, $5x$, and $8x$.
Height $(h)= \sqrt{(5x)^2 - (4x)^2} = 3x$
The area of an isosceles triangle $=\frac{1}{2} \times \text{b} \times \text{h}$
⇒ $12 = \frac{1}{2} \times 8x \times 3x$
⇒ $24=24x^2$
⇒ $x^2=1$
⇒ $x = 1$
The length of the equal sides of the triangle $=5x = 5 \times 1 = 5\;\mathrm{cm}$
The length of the equal sides $=5\;\mathrm{cm}$
Hence, the correct answer is $5$.
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