Question

The sides of a $\Delta$ are in A.P. and its area is $\frac{3}{5}\times$ (area of an equilateral triangle of the same perimeter). Find the ratio of its sides.

• A. $a:b:c::3:5:7$
• B. $a:b:c::4:5:7$
• C. $a:b:c::5:6:7$
• D. None of these

Answer 1

The correct option is A $a:b:c::3:5:7$

Let the sides of triangle $a,b,c$ which are in AP be $x-d,x,x+d$
$\Rightarrow 2s=3x$
$\Rightarrow A=\frac{3}{5}\frac{\sqrt{3}}{4}{x}^2=\frac{3\sqrt{3}}{20}{x}^2$ ....(1)
$\Rightarrow A^2=\frac{3x}{2}(\frac{x}{2}+d)\frac{x}{2}(\frac{x}{2}-d)$
$\Rightarrow \frac{27}{400}{x}^4=\frac{3}{4}{x}^2(\frac{x^2}{4}-d^2)$
$\Rightarrow \frac{x^2}{4}-d^2=\frac{9}{100}{x}^2$
$\Rightarrow d=\frac{2x}{5}$
$\Rightarrow$ sides are $\frac{3x}{5},x,\frac{7x}{5}$
$\therefore a:b:c=3:5:7$