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In the given figure, O is the center of the circle. PT and PQ are tangents to the circle from an exterior point P. If angle TPQ = 70°, find the angle TRQ.

In the given figure, O is the center of the circle. PT and PQ are tangents to the circle from an exterior point P. If angle TPQ = 70°, find the angle TRQ. Model Paper 2, Concise Mathematics Solutions ICSE Class 10.

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Answer

Join OT and OQ.

In the given figure, O is the center of the circle. PT and PQ are tangents to the circle from an exterior point P. If angle TPQ = 70°, find the angle TRQ. Model Paper 2, Concise Mathematics Solutions ICSE Class 10.

Since, line from the center to the point of contact of tangent are perpendicular.

OT and OQ are perpendicular to PT and PQ.

∴ ∠OQP = 90° and ∠OTP = 90°

In quadrilateral OTPQ,

⇒ ∠OTP + ∠TPQ + ∠OQP + ∠TOQ = 360°

⇒ 90° + 70° + 90° + ∠TOQ = 360°

⇒ ∠TOQ = 360° - 90° - 90° - 70°

⇒ ∠TOQ = 360° - 250° = 110°.

We know that,

Angle subtended by an arc on the center is twice the angle subtended by it on any other point on the circumference.

⇒ ∠TOQ = 2∠TRQ

⇒ 2∠TRQ = 110°

⇒ ∠TRQ = 110°2\dfrac{110°}{2} = 55°.

Hence, ∠TRQ = 55°.

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