Question

Show that the angle in a semi circle in a right angle

Answer 1

REF.Image

Given : A circle with center at O.

PQ is the diameter of circle.

Subtending $\angle PAQ$ at point A on circle.

To prove : $\angle PAQ={90}^{\circ }$

Proof :

Now,

POQ is a straight line passing throgh center O.

$\therefore$ Angle subtended by arc PQ at O is

$\angle POQ={180}^{\circ }$ ______ (1)

Thus,

$\angle POQ=2\angle PAQ$

$\Rightarrow \frac{\angle POQ}{2}=\angle PAQ$

$\Rightarrow \frac{{180}^{\circ }}{2}=\angle PAQ$

$\Rightarrow \angle PAQ={90}^{\circ }$

Hence, proved.