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In the given figure, PQ is a diameter of a circle with centre O and PT is a tangent at P. QT meets the circle at R. If ∠POR = 72°, find ∠PTR.

In the given figure, PQ is a diameter of a circle with centre O and PT is a tangent at P. QT meets the circle at R. If ∠POR = 72°, find ∠PTR. Tangent Properties of Circles, RSA Mathematics Solutions ICSE Class 10.

Circles

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Answer

Arc PR subtends ∠POR at center and ∠PQR on the remaining part of the circle.

⇒ ∠PQR = 12\dfrac{1}{2} ∠POR

⇒ ∠PQR = 72°2\dfrac{72°}{2}

⇒ ∠PQR = 36°.

We know that,

The tangent at any point of a circle and the radius through this point are perpendicular to each other.

∠QPT = 90°

In triangle QPT,

⇒ ∠QPT + ∠PQR + ∠PTR = 180°

⇒ 90° + 36° + ∠PTR = 180°

⇒ 126° + ∠PTR = 180°

⇒ ∠PTR = 180° - 126°

⇒ ∠PTR = 54°.

Hence, ∠PTR = 54°.

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