Question

In figure, the side QR of $\Delta PQR$ is produced to a point S. If the bisectors of $\angle PQR$ and $\angle PRS$ meet at point T, and $\angle QRT=\frac{1}{k}\angle QPR$, then find $k$.

• A. $4$
• B. $3$
• C. $2$
• D. $1$

Answer 1

The correct option is C $2$

$\angle PRS=\angle PQR+\angle QPR$ ...(1)

Now, $\angle QTR+\angle TQR=\angle TRS$

$\Rightarrow \angle QTR+\frac{1}{2}\times \angle PQR=\frac{1}{2}\angle PRS$

$\Rightarrow 2\angle QTR+\angle PQR=\angle PRS$ .....(2)

From (1) and (2),

$2\angle QRT+\angle PQR=\angle PQR+\angle QPR$

$\Rightarrow 2\angle QTR=\angle QPR$

$\Rightarrow QRT=\frac{1}{2}\angle QPR$

comparing it with $\angle QRT=\frac{1}{k}\angle QPR$

we get, $k=2$.