Question

$PQ$ and $QR$ are tangents to the circle at P and R respectively. Find the value of x

• A. ${25}^{\circ }$
• B. ${30}^{\circ }$
• C. ${55}^{\circ }$
• D. ${40}^{\circ }$

Answer 1

The correct option is B ${30}^{\circ }$

Since PQ is a tangent

$OP\perp PQ$

$⟹\angle OPQ=90$

$\angle OPR+\angle RPQ=90$

$\angle OPR=90-65=25$

In $△OPR$

$OP=OR$

$⟹\angle OPR=\angle ORP=25$

$\angle POR=180-\angle OPR-\angle ORP=180–50-130$

$\angle SOP+\angle POR=\angle SOR=360$

$\angle SOP=360–110–130=120$

In $△SOP$

$OS=OP$

$\angle OSP=\angle OPS$

$\angle OSP+\angle OPS=\angle SOP=180$

$2x=180-120$

$x={30}^{\circ }$