Question

In a $△$ PQR, P is the largest angle and cos P = $\frac{1}{3}$.
Further in circle of the triangle touches the sides
PQ,QR and RP at N ,L and M respectively, such
th a t the lengths of P N , QL and RM a re 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)

PLAN Whenever cosine of angle and sides are given or to find out, we
should always use Cosine law.
i,e., cos A = $\frac{b^2+c^2-a^2}{2bc},cosB=\frac{a^2+c^2-b^2}{2ac}$
and cos C = $\frac{a^2+b^2-c^2}{2ab}$
$\therefore cosP=\frac{b^2+c^2-a^2}{2bc}$
$\Rightarrow \frac{1}{3}=\frac{(2n+4{)}^2+(2n+2{)}^2-(2n+6{)}^2}{2(2n+4)(2n+2)}$
$[\because cosp=\frac{1}{3},given]$
$\Rightarrow \frac{4n^2-16}{8(n+1)(n+2)}=\frac{1}{3}$
$\Rightarrow \frac{n^2-4}{2(n+1)(n+2)}=\frac{1}{3}$
$\Rightarrow \frac{(n-2)}{2(n+1)}\frac{1}{3}$
$\Rightarrow 3n-6=2n+2\Rightarrow n=8$.
$\therefore$ Sides are (2n + 2), (2n + 4), (2n + 6), i.e, 18,20,22