CBSE2026Class Class 10 · Mathematics1 Marks · Mcq✅ Verified
PQ is tangent to a circle with centre O at the point P, SP is a diameter and R is a point on the circle such that SR produced meets the tangent at T. If $\angle POR = 65^\circ$, then $m\angle PTR$ is
A. $65^\circ$
B. $58.5^\circ$
C. $57.5^\circ$
D. $45^\circ$
✅ Answer & Solution
✅ Correct Answer: C
Step 1: OP is a radius and PQ is a tangent at P, so $\angle OPT=90^\circ$, i.e. $\angle SPT=90^\circ$.
Step 2: $\angle POR=65^\circ$ is the central angle standing on arc PR.
Step 3: $\angle PSR$ is the inscribed angle on the same arc PR, so
$$\angle PSR=\frac{1}{2}\angle POR=\frac{65^\circ}{2}=32.5^\circ$$
Step 4: In $\triangle SPT$, angle sum $=180^\circ$:
$$\angle PTS=180^\circ-90^\circ-32.5^\circ=57.5^\circ$$
Step 5: T, R, S are collinear, so $\angle PTR=\angle PTS=57.5^\circ$.
Answer: (C) $57.5^\circ$
