Find the percentage increased in area of triangle if its each side is doubled(explain with figure) - Maths - Heron\s Formula

Question

Find the percentage increased in area of triangle if its each
side is doubled(explain with figure)

Harleen Birdi · Accepted Answer

Let a, b, c be the sides of the old triangle and s be its semi-perimeter.

$Then, s = \frac{1}{2} (a + b + c)$

The sides of the new triangle are 2a, 2b and 2c. Let s' be its semi-perimeter. Then,

$s' = \frac{1}{2} \times (2 a + 2 b + 2 c) = a + b + c = 2 s$

Let Δ and Δ' be the areas of the old and new triangles respectively. Then,

$Δ = \sqrt{s (s - a) (s - b) (s - c)} and Δ' = \sqrt{s' (s' - 2 a) (s' - 2 b) (s' - 2 c)} \Rightarrow Δ' = \sqrt{2 s (2 s - 2 a) (2 s - 2 b) (2 s - 2 c)} \Rightarrow Δ' = \sqrt{s (s - a) (s - b) (s - c)} = 4 Δ$

∴ Increase in the area of the triangle = Δ' – Δ = 4Δ – Δ = 3Δ.

Hence, percentage increase in area = $(\frac{3 Δ}{Δ} \times 100) = 300 %$ .

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