Question

In the given figure, 3 secants are drawn from point P, intersects the circle at Q, M, and R respectively. What is the value of $\angle QXR$?

• A. ${100}^{\circ }$
• B. ${20}^{\circ }$
• C. ${90}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

The correct option is A

${100}^{\circ }$

In $\Delta QMP$,

$\angle MQP+\angle QMP+\angle MPQ={180}^{\circ }$ [Sum of angles of a trianlge is ${180}^{\circ }$]

$\Rightarrow \angle MQP+{30}^{\circ }+{40}^{\circ }={180}^{\circ }\angle MQP+{70}^{\circ }={180}^{\circ }\angle MQP={180}^{\circ }-{70}^{\circ }\angle MQP={110}^{\circ }=\angle MQX$

In the quadrilateral QXMA, sum of interior angles of a quadrialateral = ${360}^{\circ }$

$\angle MQX+\angle QXZ+\angle XZM+\angle ZMQ={360}^{\circ }$

${110}^{\circ }+\angle QXZ+{120}^{\circ }+{30}^{\circ }={360}^{\circ }{260}^{\circ }+\angle QXZ={360}^{\circ }\angle QXZ={360}^{\circ }-{260}^{\circ }\angle =QXZ=\angle QXR={100}^{\circ }$