Question

If each side of an equilateral triangle is increased by 2 cm., then it's area is increased by 3root 3 cm square. Find the length of each side and its area

Answer 1

Let  the  initial side of  Δ be  a₁ , area A₁ and  final side  be a₂, area A₂

Area  of  triangle, A₁ = √3a₁²/4

Increasing  its  area  by  2 cm,

New  side, a₂ = a₁ +2

New  area, A₂ = √3a₂²/4

 but  A₂ = A₁ +3√3, as t's area is increased by 3√3 meter square.

A₁ +3√3 = √3a₂²/4

√3a₁²/4 + 3√3 = √3(a₁+2)²/4

$\frac{\sqrt{3}a_1^{2}}{4}+3\sqrt{3}=\frac{\sqrt{3}(a_1+2)^{2}}{4}\\   \\\frac{a_1^{2}}{4}+3=\frac{(a_1+2)^{2}}{4}\\\\\frac{a_1^{2}}{4}+3=\frac{a_1^{2}+4+2\times2\times a_1}{4}\\\\\frac{a_1^{2}}{4}+3=\frac{a_1^{2}}{4}+\frac{4}{4}+\frac{4a_1}{4}\\\\3=1+a_1\\\\{a_1}=3-1\\\\a_1=2\\\\Area\;of\;triangle,\;\;A_1=\frac{\sqrt{3}.2^{2}}{4}=\sqrt{3}$