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# Tangents Ap and Aq Are Drawn to a Circle, with Centre O, from an Exterior Point A. Prove That: Paq = 2∠Opq - ICSE Class 10 - Mathematics

ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

#### Question

Tangents AP and AQ are drawn to a circle, with centre O, from an exterior point A. prove that: PAQ = 2∠OPQ

#### Solution

∠OPA = ∠ OQA = 90º

(∵ OP ⊥ PA and  OQ ⊥ QA)

∴ ∠POQ + ∠PAQ +90° +90°=360°

⇒ ∠POQ +∠PAQ =360° -180°=180°   ........................(i)

In triangle OPQ,
OP = OQ (Radii of the same circle)
∴ OPQ = ∠OQP
But

∠POQ + ∠ OPQ + ∠ OQP =180°

⇒ ∠POQ + ∠OPQ +∠OPQ = 180°

⇒ ∠POQ + 2 ∠OPQ = 180°   ...............................(ii)

From (i) and (ii)
∠POQ + ∠PAQ = ∠POQ + 2 OPQ
⇒ ∠ PAQ  = 2 ∠ OPQ

Is there an error in this question or solution?

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Solution Tangents Ap and Aq Are Drawn to a Circle, with Centre O, from an Exterior Point A. Prove That: Paq = 2∠Opq Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
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