If two tangents AP and AQ are drawn

SSC CHSL Additional Result announced on 2nd September 2022. This is with reference to the Tier I examination 2021. The detailed notification for SSC CHSL 2022 will be released on 5th November 2022. Candidates who have completed Higher Secondary (10+2) can appear for this exam for recruitment to various posts like Postal Assistant, Lower Divisional Clerks, Court Clerk, Sorting Assistants, Data Entry Operators, etc. The commission has also announced a total 9838 vacancies (tentative) for the posts of Postal Assistant and Sorting Assistant this year. Candidates can check the SSC CHSL result details here.

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

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

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

  Is there an error in this question or solution?

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

Open in App