Question

In a triangle $PQR$, $P$ is the largest angle and $\cos P=1/3$. 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. $16,18,20$
• B. $18,20,22$
• C. $24,26,28$
• D. $22,24,26$

Answer 1

The correct option is B $18,20,22$

$\cos P=\frac{(2x+2{)}^2+(2x+4{)}^2-(2x+6{)}^2}{2(2x+2)(2x+4)}$

$\Rightarrow \frac{1}{3}=\frac{4x^2-16}{2(4x^2+12x+8)}$

$\Rightarrow 8x^2+24x+16=12x^2-48$

$\Rightarrow 4x^2-24x-64=0$

$\Rightarrow x^2-6x-16=0$

$\Rightarrow (x-8)(x-2)=0$

$\Rightarrow x=8$

So, the sides of triangle are 18,20,22