Question

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

Answer 1

Let AB be a chord of a circle with centre O, and let AD and BD be the tangents at A and B respectively.

Suppose OD meets AB at C.

We have to prove that ∠DAC = ∠DBC.

We know that, the line segment joining the centre to the external point, bisects the angle between two tangents.

So, ∠ADC=∠BDC ...(i)

In △DCA and △DCB, we have

DA = DB [Tangents from an external point are equal]

∠ADC = ∠BDC [From (i)]

DC=DC [Common]

∴ ∆DCA ≅ ∆DCB [By SAS congruency rule]

⇒∠DAC = ∠DBC [BY C.P.CT]