Find the percentage increase in the area of the triangle if its each side is doubled.

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Solution

Let triangle be with sides a,b,c
$S_{1} = \frac{a + b + c}{2}; A r e a = \sqrt{S_{1} (S_{1} - a) (S_{1} - b) (S_{1} - c)} = A_{1}$
Sides doubled $\Rightarrow S_{2} = \frac{2 a + 2 b + 2 c}{2} = a + b + c$
$\Rightarrow S_{2} = 2 S_{1}$
$∴ A_{2} = \sqrt{S_{2} (S_{2} - 2 a) (S_{2} - 2 b) (S_{2} - 2 c)}$
$= \sqrt{2 S_{1} (2 S_{1} - 2 a) (2 S_{1} - 2 b) (2 S_{1} - 2 c)} = \sqrt{2 \times 2 \times 2 \times 2 \times \times S_{1} (S_{1} - a) (S_{1} - b) (S_{1} - c)} = 2 \times 2 \times \sqrt{S_{1} (S_{1} - a) (S_{1} - b) (S_{1} - c)} A_{2} = 4 A_{1} ∴ % a g e i n c r e a s e = \frac{A_{2} - A_{1}}{A_{1}} \times 100 = \frac{4 A_{1} - A_{1}}{A_{1}} \times 100 = 300 %$ .

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