# The sides of a triangle are in A.P. and its area is 3/5 th the area of an equilateral triangle of same perimeter.

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

(a) 1 : 2 :√7

(b) 2 : 3 : 5

(c) 1 : 6 : 7

(d) 3 : 5 : 7

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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.