Q.
If the sides of a traingle are three consecutive natural numbers and its largest angle is twice the smallest one, then the area (in sq. units) of that triangle is
Let three sides are x−1,x,x+1
Given, largest angle is twice the smallest angle ∴cosθ=2x(x+1)(x+1)2+(x)2−(x−1)2 =2x(x+1)x2+2x+1+x2−x2+2x−1 =2x(x+1)x2+4x=2(x+1)x+4
By Sine rule, sin2θx+1=sinθx−1 ⇒2sinθcosθx+1=sinθx−1 ⇒cosθ=2(x−1)x+1
From Eqs. (i) and (ii), 2(x−1)x+1=2(x+1)x+4 ⇒x=5 ∴ Sides are 4,5,6 Δ=21215(215−4)(215−5)(215−6) =4115×7×5×3=4157