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

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Last updated date: 25th Jun 2024
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Answer
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Hint:To find the percentage increase of the area of the triangle we first find the area of the triangle by using the area of the triangle as:
Area of the triangle when all sides are same \[=\dfrac{\sqrt{3}}{4}side{{s}^{2}}\]
The triangle is taken as an equilateral triangle as each side is doubled and for every side to increase equally all the sides are to be the same.

Complete step by step solution:
Let us assume the area of the triangle as \[=\dfrac{\sqrt{3}}{4}side{{s}^{2}}\] which is the area of the equilateral triangle, an equilateral triangle is a triangle which divides the triangle into half each producing two right angle triangles hence, let us assume the value of the sides as \[x\].
\[\Rightarrow \dfrac{\sqrt{3}}{4}side{{s}^{2}}\]
\[\Rightarrow \dfrac{\sqrt{3}}{4}\times x\times x\]
\[\Rightarrow \dfrac{\sqrt{3}}{4}{{x}^{2}}\]
Now to find the area of the triangle after the sides are increased is by \[2x\] and we place this value in the area of the triangle as:
\[\Rightarrow \dfrac{\sqrt{3}}{4}side{{s}^{2}}\]
\[\Rightarrow \dfrac{\sqrt{3}}{4}\times 2x\times 2x\]
\[\Rightarrow \sqrt{3}{{x}^{2}}\]
Hence, the area when the sides are doubled is given as\[\sqrt{3}{{x}^{2}}\]
Now to find the increase in the area from the original, we get the increase in area as:
Increased Area\[-\] Original Area\[=\] Difference in the Area
Placing the area in the above formula, we get the difference as:
\[\Rightarrow \sqrt{3}{{x}^{2}}-\dfrac{\sqrt{3}}{4}{{x}^{2}}=\dfrac{3\sqrt{3}}{4}{{x}^{2}}\]
Now forming a percentage for the increased value as:
\[\Rightarrow \dfrac{\dfrac{3\sqrt{3}}{4}{{x}^{2}}}{\dfrac{\sqrt{3}}{4}{{x}^{2}}}\times 100\]
\[\Rightarrow 300%\]

Note: Students may go wrong if they use any other triangle other than equilateral triangle such as scalene triangle, right angle triangle or isosceles triangle as the increase in area has to be equal from each side of the triangle therefore, we use the equilateral triangle.