Question

P and Q are two points on a circle with centre at O. R is a point on the minor arc of the circle between the points P and Q. he tangents to the circle at the pints p and Q meet each other at the point S. If $\angle PSQ=20,\angle PRQ=?$

• A. 80
• B. 200
• C. 160
• D. 100

Answer 1

The correct option is D 100

R.E.F image

In $OPSQ$

$\angle 0+\angle P+\angle S+\angle Q={360}^{\circ }$

$\Rightarrow \angle O+{90}^{\circ }+{20}^{\circ }+{90}^{\circ }={360}^{\circ }$

$\Rightarrow \angle O={160}^{\circ }$

Angle subtended by PQ at center is twice

that of subtended at circumference

$\Rightarrow \angle O=2\angle PMQ$

$\Rightarrow {160}^{\circ }=2\angle PMQ$

$\Rightarrow \angle PMQ={80}^{\circ }$

In PMQR, Sum of opposite angles $={180}^{\circ }$ (PMQR is cyclic)

$\Rightarrow \angle M+\angle R={180}^{\circ }$

$\Rightarrow {80}^{\circ }+\angle R={180}^{\circ }$

$\Rightarrow \angle R={100}^{\circ }$