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in Co-ordinate geometry by (63.5k points)

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle. 

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Let ABC be the triangle such that the lengths of its sides CA, AB and BC are x – 1, x and x + 1 respectively where x ∈ N and x > 1. 

Let ∠B = α be the smallest angle and ∠A = 2α be the largest angle.

Then, by sine rule, we have

using cosine law

⇒ cos α = (x + 4)/2(x - 1)

From Eqs. (i) and (ii), we get

(x + 1)/2(x - 1) = (x + 4)8/2(x + 1)

⇒ (x + 1)2 = (x + 4) (x – 1) 

⇒ x2 + 2x + 1 = x2 + 3x – 4⇒ x = 5

Hence, the lengths of the sides of the triangle are 4, 5 and 6 unit.

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