Question

The sides of a triangle are in $AP$ and its area is $\frac{3}{5}$ times the area of an equilateral triangle of the same perimeter.

Then the ratio of the sides is?

• A. $1:2:3$
• B. $3:5:7$
• C. $1:3:5$
• D. none of these

Answer 1

Answer: (B)

$3:5:7$

Let the sides be $a,a+d,a+2d$
Perimeter of triangle $2s=3a+3d=3p$, where p is the side of equilateral triangle with same perimeter
$\Rightarrow s=\frac{3}{2}(a+d)andp=a+d$
Area of the triangle is $\frac{3}{5}$th that of the equilateral triangle with same perimeter
$\Rightarrow \sqrt{\frac{3}{2}(a+d)\frac{1}{2}(a+3d)\frac{1}{2}(a+d)\frac{1}{2}(a-d)}=\frac{3}{5}\ast \frac{\sqrt{3}}{4}{(a+d)}^2\Rightarrow 25(a^2-3d^2+2ad)=9(a^2+d^2+2ad)\Rightarrow 4a^2-21d^2+8ad=0\Rightarrow d=\frac{2a}{3}or-\frac{2a}{7}$
In both cases the ratio of sides is $3:5:7$