Question

Prove that the tangents drawn at the ends of a chord of circle make equal angles with the chord.

Answer 1

Given: A circle with centre O, PA and PB are tangents drawn at the ends A and B on chord AB.

To prove: ∠PAB = ∠PBA
Construction: Join OA and OB.
Proof: In DOAB we have
OA = OB     …(i)

[Radii of the same circle]

∠2 = ∠1      …(ii) 

[Angles opposite to equal sides of a D]
Also (∠2 + ∠3 = ∠1 + ∠4)      ...(iii) 

[Both 90° as Radius ⊥ Tangent]
Subtracting (ii) from (iii), we have

∠3 = ∠4 

⇒ ∠PAB = ∠PBA
Hence proved.