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In the figure given below, O is the center of the circle and SP is a tangent. If ∠SRT = 65°, find the values of x, y and z.

In the figure, O is the center of the circle and SP is a tangent. If ∠SRT = 65°, find the values of x, y and z. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

Circles

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Answer

As, SP is tangent to the circle.

∴ ∠TSR = 90°.

In △TSR,

⇒ ∠TSR + ∠STR + ∠SRT = 180° [By angle sum property of triangle]

⇒ 90° + x + 65° = 180°

⇒ x = 180° - 155° = 25°.

We know that,

Angle which an arc subtends at the center is double that which it subtends at any point on the remaining part of the circumference.

∴ ∠SOQ = 2∠STQ

⇒ y = 2x = 2(25°) = 50°.

In △OSP,

⇒ ∠OSP + ∠SOP + ∠SPO = 180° [By angle sum property of triangle]

⇒ 90° + y + z = 180°

⇒ 90° + 50° + z = 180°

⇒ z = 180° - 140° = 40°.

Hence, x = 25°, y = 50° and z = 40°.

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