    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 length 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 option is D 22The largest side is QR. PM=PN=2kRM=RL=2k+4QL=QN=2k+2 The sides of the triangle is given as, QR=4k+6RP=4k+4QP=4k+2cosP=(PQ)2+(PR)2−(QR)22(PQ)(PR) ⇒13=(4k+2)2+(4k+4)2−(4k+6)22(4k+2)(4k+4)⇒13=(2k+1)2+4(k+1)2−(2k+3)24(2k+1)(k+1)⇒13=4(k+1)2−(4k+4)(2)4(2k+1)(k+1) Further simplify for the value of k. 13=(k+1)2−2(k+1)(2k+1)(k+1)⇒(k+1)(2k+1)=3(k−1)(k+1)⇒k=−1 & 4 Consider the positive value of k ∴QR=4×4+6=22∴RP=4×4+4=20∴QR=4×4+2=18  Suggest Corrections  0      Similar questions
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