In a triangle PQR, P is the largest angle and cos P = \frac{1}{3} . Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is(are)

Engineering

Mathematics

Basic Rules of Properties of Triangle

Question

In a triangle PQR, P is the largest angle and cos P = $\frac{1}{3}$ . Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is(are)

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cos p = $\frac{1}{3}$ = (4n + 2)² + (4n + 4)² – (4n + 6)²

2(4n + 2)(4n + 4)

$\frac{1}{3} = \frac{16 (n^{2} - 1)}{16 (2 n + 1) (n + 1)}$

⇒ (2n + 1)(n +1) = 3n² = 3

⇒ n² – 3n – 4 = 0

⇒ (n – 4)(n +1) = 0

$∴$ n = 4

So sides are 18, 20, 22

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