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Question

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

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Solution

Let a, b, c be the sides of the old triangle and s be its semi-perimeter. Then,
s=12(a+b+c)
The sides of the new triangle are 2a,2b and 2c
.Let s' be its semi-perimeter. Then,
s=12×(2a+2b+2c)
=a+b+c=2s
Let Δ and Δ be the areas of the old and new traiangles respectively. Then,
Δ=s(sa)(sb)(sc) and
Δ=s(s2a)(s2b)(s2c)
Δ=2s(2s2a)(2s2b)(2s2c) [ s=2s]
Δ=4s(sa)(s)(sc)=4Δ
Increase in the are of the triangle
=ΔΔ=4ΔΔ=3Δ
Hence, percentage increase in area
=(3ΔΔ×100)=300%

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