In a ∆ P Q R , P is the largest angle and cos P = 1 3 . Further the incircle of the triangle touches the sides P Q , Q R and R P at N , L and M respectively, such that the lengths of P N , Q L and R M are consecutive even integers. Th

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In a &#8710; P Q R , P is the largest angle and cos P = 1 3 . Further the incircle of the triangle touches the sides P Q , Q R and R P at N , L and M respectively, such that the lengths of P N , Q L and R M are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

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Correct answer: B. 18

Consider the following figure: We assumed the value of sides in terms of n , which satisfies our given condition. Now applying cosine rule, cos P = 2 n + 2 2 + 2 n + 4 2 - 2 n + 6 2 2 2 n + 2 2 n + 4 = 4 n 2 - 16 8 n + 1 n + 2 = 1 3 = n 2 - 4 2 n + 1 n + 2 = 1 3 &#8658; n - 2 2 n + 1 = 1 3 &#8658; 3 n - 6 = 2 n + 2 &#8658; n = 8 Length of the sides will be 18 , &#160; 20 &#160; and 22 .

In a ∆ P Q R , P is the largest angle and cos P = 1 3 . Further the incircle of the triangle touches the sides P Q , Q R and R P at N , L and M respectively, such that the lengths of P N , Q L and R M are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

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