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In the following figure, Q is the centre of a circle and PM, PN are tangent segments to the circle. If ∠MPN = 50°, find ∠MQN. - Geometry

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In the following figure, Q is the centre of a circle and PM, PN are tangent segments to the circle. If ∠MPN = 50°, find ∠MQN.

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Solution

Seg PM and seg PN are tangents to the circle and seg QM and seg QN are the radii from the points of contacts.

m∠PMQ = m∠PNQ = 90°                  ... (Tangent is perpendicular to the radius) ... (1)

The sum of the measures of the angles of a quadrilateral is 360°.

m∠MPN + m∠PMQ + m∠MQN + m∠PNQ = 360°

50° + 90° + m∠MQN + 90° = 360°

230° + m∠MQN = 360°

m∠MQN = 360° – 230° = 130°                     ... [From (1)]

Concept: Number of Tangents from a Point on a Circle
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