Question

In the figure P , Q and R are the points where the incircle of $△ABC$ touches the sides :

Using the information given , $\angle QRP$ = ____

• A. ${60}^{\circ }$
• B. ${70}^{\circ }$
• C. ${50}^{\circ }$
• D. ${45}^{\circ }$

Answer 1

The correct option is C

${50}^{\circ }$

Join IQ and IP.

$\angle ACB$ = ${180}^{\circ }$ - (${40}^{\circ }+{60}^{\circ }$) = ${180}^{\circ }+{100}^{\circ }$ = ${80}^{\circ }$...... (Sum of angles of a $△$)

$\angle IQC$ = $\angle IPC$ = ${90}^{\circ }$ ...........(Tangent is perpendicular to the radius at the point of contact)

$\angle QIP$ = ${180}^{\circ }$ - ${80}^{\circ }$ = ${100}^{\circ }$ ..........(Sum of angles of a quadrilateral)

$\angle QRP$ = $\frac{1}{2}\angle QIP$ = $\frac{1}{2}\times {100}^{\circ }$ = ${50}^{\circ }$ ........(Angles at the circumference is half the angle at the centre)