Question

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

Answer 1

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

To prove:- $\angle PAB=\angle PBA$

Construction:- Join $OA$ and $OB$

Proof:- In $△AOB$, we have

$OA=OB\left(\text{Radii of the same circle}\right)$

$\angle OAB=\angle OBA.....\left(1\right)\left(\text{Angles opposite to equal sides}\right)$

$\angle OAP=\angle OBP=90(\because \text{Radius}\perp \text{Tangent})$

$\Rightarrow \angle OAB+\angle PAB=\angle OBA+\angle PBA$

$\Rightarrow \angle OAB+\angle PAB=\angle OAB+\angle PBA\left(\text{From}\left(1\right)\right)$

$\Rightarrow \angle PAB=\angle PBA$

Hence proved.