Question

In the given figure, AB is diameter of the circle with centre O. AQ, BP and PRQ are tangents. Prove that OP and OQ are perpendicular to each other.

Answer 1

In the above figure first of all join $OR$

Now, from the figure it is seen that

$△OBP\cong △ORP$

Then, $\angle BOP=\angle ROP=x$

Now, from the figure it is also seen that

$△OAQ\cong △ORQ$

Since $AOB$ is straight line,

$\angle AOB={180}^o$

$\therefore x+x+y+y={180}^o$

$\therefore 2x+2y={180}^o$

$\therefore 2(x+y)={180}^o$

$\therefore (x+y)={90}^o$

$\therefore \angle POQ={90}^o$

$\to$ Hence $OP$ and $OQ$ are perpendicular to each other.