Question

In a $\Delta PQR$, P is the largest angle and $cosP=\frac{1}{3}$.
Further 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)

22

$\therefore cosP=\frac{b^2+c^2-a^2}{2bc}\Rightarrow \frac{1}{3}=\frac{(2n+4{)}^2+(2n+2{)}^2-(2n+6{)}^2}{2(2n+4)(2n+2)}[\because cosP=\frac{1}{3},given]\Rightarrow \frac{4n^2-16}{8(n+1)(n+2)}=\frac{1}{3}\Rightarrow \frac{n^2-4}{2(n+1)(n+2)}=\frac{1}{3}\Rightarrow \frac{(n-2)}{2(n+1)}=\frac{1}{3}\Rightarrow 3n-6=2n+2\Rightarrow n=8$
$\therefore$ Sides are (2n+2), (2n+4), (2n+6), i.e. 18, 20, 22.