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Question

In a triangle PQR, P is the largest angle and cosP=13 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)

A
16
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B
18
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C
24
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D
22
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Solution

The correct options are
A 18
D 22
Let sa=2k2,sb=2k,sc=2k+2, kI, k>1

Adding we get,

s=6k

So a=4k+2, b=4k, c=4k2

Now, cosP=13

b2+c2a22bc=133[(4k)2+(4k2)2(4k+2)2]=2×4k(4k2)

3[16k24(4k)×2]=8k(4k2)

48k296k=32k216k

16k2=80kk=5

So, sides are 22,20,18


102375_31963_ans.PNG

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