Mathematics
If PQ is a tangent to the circle at R; calculate :
(i) ∠PRS,
(ii) ∠ROT.
Given, O is the center of the circle and angle TRQ = 30°.
Circles
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Answer
(i) Since, ST passes through O, so ST is the diameter of the circle.
We know that,
Angle in a semi-circle is a right angle.
∴ ∠SRT = 90°.
Since, PQ is a straight line.
∴ ∠PRS + ∠SRT + ∠TRQ = 180°
⇒ ∠PRS + 90° + 30° = 180°
⇒ ∠PRS + 120° = 180°
⇒ ∠PRS = 180° - 120°
⇒ ∠PRS = 60°.
Hence, ∠PRS = 60°.
(ii) We know that,
The angle between a tangent and chord through the point of contact is equal to an angle in the alternate segment.
∠TSR = ∠TRQ = 30°.
Since, angle subtended by a segment at the center is double the angle suspended at the circumference.
∠ROT = 2∠TSR = 2 × 30° = 60°.
Hence, ∠ROT = 60°.
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