Answer : (d) 3 : 5 : 7
Let the sides of the triangle be a – d, a, a + d.
Perimeter of the triangle = a – d + a + a + d = 3a
∴ Each side of the equilateral triangle = \(\frac{3a}{3}\) = a
∴ Area of equilateral triangle = \(\frac{\sqrt{3}}{4}a^2\)
Area of the given ∆ = \(\sqrt{s(s-(a-d)(s-a)(s-(a+d))}\) where s = \(\frac{3a}{2}\)
Given, \(\sqrt{\frac{3}{4}a^2(\frac{1}{4}a^2 -d^2)}\) = \(\frac{3}{5} \times\) \(\frac{\sqrt{3}}{4}a^2\)
⇒ \(\frac{25a^2 - 9a^2}{400}\) = \(\frac{1}{4}d^2\)
⇒ \(\frac{16a^2}{400}\) = \(\frac{1}{4}d^2\)
⇒ \(\frac{a^2}{d^2}\) = \(\frac{400}{64}\)
⇒ \(\frac{a}{d}\) = \(\frac{20}{8}\) = k (say)
∴ (a – d) : a : (a + d) = (20k – 8k) : 20k : (20k + 8k)
= 12k : 20k : 28k
= 3 : 5 : 7.
