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)

• A. 16
• B. 18
• C. 24
• D. 22

Answer 1

Answer: (B), (D)

18
22

Let $s-a=2k-2,s-b=2k,s-c=2k+2,k\in I,k>1$

Adding we get,

$s=6k$

So $a=4k+2,b=4k,c=4k-2$

Now, $\cos P=\frac{1}{3}$

$\Rightarrow \frac{b^2+c^2-a^2}{2bc}=\frac{1}{3}\Rightarrow 3\left[\right(4k{)}^2+(4k-2{)}^2-(4k+2{)}^2]=2\times 4k(4k-2)$

$\Rightarrow 3[16k^2-4(4k)\times 2]=8k(4k-2)$

$\Rightarrow 48k^2-96k=32k^2-16k$

$\Rightarrow 16k^2=80k\Rightarrow k=5$

So, sides are $22,20,18$