In a triangle PQR, P is the largest angle and cos P = 
. 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)
Text Solution
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,
Let
s - a = 2k - 2, s - b = 2k, s - c = 2k + 2, k 
I, k > 1
Adding we get,
s = 6k
So, a = 4k + 2, b = 4k, c = 4k - 2
Now, cos P = 
3 [(4k) 2 + (4k - 2) 2 - (4k + 2) 2 ] = 2 x 4k (4k - 2)
3 [16k 2 - 4 (4k) x 2] = 8k (4k - 2)
48k 2 - 96k = 32k 2 - 16k
16k 2 = 80k 
k = 5
So, sides are 22, 20, 18
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