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Pq is a Tangent Drawn from an External Point P to a Circle with Centre O, Qor is the Diameter of the Circle. If ∠Por = 120°, What is the Measure of ∠Opq? - Mathematics

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Question

PQ is a tangent drawn from an external point P to a circle with centre O, QOR is the diameter of the circle. If ∠POR = 120°, what is the measure of ∠OPQ?

Solution

Given: PQ is the tangent to the circle with centre O. QOR is the diameter.

\[\angle POR = 120^o\]

In\[∆ POQ\]

\[\angle PQO = 90^o\]                                              (tangent at any point of a circle is perpendicular to the radius through the point of contact)
\[\angle OPQ + \angle PQO = \angle POR\]                           (Exterior angle = sum of interior opposite angles)
\[\angle OPQ + 90^o = 120^0\]
\[ \Rightarrow \angle OPQ = 120^o - 90^o = 30^o\]

 

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