Question:
If two tangents are drawn to a circle from an external point,show that they subtend equal angles at the centre.
Solution:
Given : A circle with centre O and a point A outside it. Also, AP and AQ are the two tangents to the circle.
:To prove: $\angle A O P=\angle A O Q$.
Proof : In $\Delta A O P$ and $\Delta A O Q$, we have :
$A P=A Q \quad$ [tangents from an external point are equal]
$O P=O Q \quad$ [radii of the same circle]
$O A=O A \quad[$ common side $]$
$\therefore \Delta A O P \cong \Delta A O Q \quad$ [by SSS - congruence]
Hence, $\angle A O P=\angle A O Q$ (c. p. c. $\mathrm{t}$ ).