Find the percentage increase in the area of a triangle if each side is increased to ^@5^@ times.


Answer:

^@ 2400 \% ^@

Step by Step Explanation:
  1. Consider a triangle ^@QRS^@ with sides ^@a, b^@ and ^@c.^@
    Let ^@ S = \dfrac { a+b+c } { 2 } ^@
    Area of triangle ^@ QRS, A_1 = \sqrt{ S(S-a)(S-b)(S-c) } ^@
  2. Increasing the side of each side by ^@5^@ times, we get a new triangle ^@XYZ.^@
    ^@XYZ^@ has sides ^@5a, 5b^@ and ^@5c.^@
    By Heron's formula,
    Area of new triangle ^@= \sqrt{ S_{1}(S_{1}-5a)(S_{1}-5b)(S_{1}-5c) }^@
    Where ^@S_1 = \dfrac { 5a + 5b + 5c } { 2 } = 5 \times \dfrac { a+b+c } { 2 }^@
    Area of ^@XYZ = \sqrt{ 5S(5S-5a)(5S-5b)(5S-5c) } ^@
    @^ = \sqrt{ 5^{4}S(S-a)(S-b)(S-c) } \\ = 5^2 \times A_1 \\ = 25 A_1 @^
  3. Percentage increase in the area of the triangle, @^ \begin{align} &= \dfrac{ \text{ Area of Triangle XYZ } - \text{ Area of Triangle QRS } } { \text{Area of Triangle QRS} } \times 100 \\ &= \dfrac{ 25 A_1 - A_1 } { A_1 } \times 100 \\ &= \dfrac{ 24 A_1 }{ A_1 } \times 100 \\ &= 2400 \end{align} @^
  4. This means the area of the triangle, ^@ A_1 ^@ is increased by ^@ 2400 \%.^@

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