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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

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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

In quadrilateral OPAQ,

`∠`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

 

 

 

 

 

 

 

 

 

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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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