Question

In the given figure,
$AB$ and $AC$ are tangents to the circle with centre $O.$
Given that $\angle BAC={70}^{\circ }$ and $P$ is a point on the minor arc $BC,$ find $\angle BPC.$

• A. ${110}^{\circ }$
• B. ${140}^{\circ }$
• C. ${125}^{\circ }$
• D. ${136}^{\circ }$

Answer 1

Answer: (C)

${125}^{\circ }$

From the given figure,
$\angle COB+\angle CAB={180}^{\circ }$

(The angle between two tangents drawn from an external pt. to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre)
$\Rightarrow$ $\angle COB={180}^{\circ }-{70}^{\circ }={110}^{\circ }$

Reflex $\angle COB={360}^{\circ }-{110}^{\circ }={250}^{\circ }$

$\therefore$ $\angle BPC=\frac{1}{2}\times$ Reflex$\angle AOB=\frac{1}{2}\times {250}^{\circ }={125}^o$

(Angle subtended by the arc at the centre is twice the angle subtended at the circle)