Question

Prove that the angle in a semicircle is a right angle.

Answer 1

Hint: Here we will be using the theorem/property of circle, the angle subtended by an arc at the center is double the angle subtended by it on any point on the remaining part of the circle as shown in figure.

Complete step-by-step answer:
According to question, the figure can be drawn as shown-

Here, AOB is a straight line passing through center O.
$\therefore $ Angle subtended by the arc AB at O is
$\angle AOB = 180^\circ $
The theorem states that the angle subtended by an arc at the center is double the angle subtended by it on any point on the remaining part of the circle.
$\therefore \angle AOB = 2\angle APB$ ($\angle AOB$ is the subtended at center which is equal to $180^\circ $ and $\angle APB$ is the angle made at any point on the circle.)
$\dfrac{{\angle AOB}}{2} = \angle APB$
$\dfrac{{180^\circ }}{2} = \angle APB$
$\angle APB = 90^\circ $
Hence, it can be said that the angle in a semicircle is a right angle.

Note- For solving problems related to angles subtended by an arc in a circle, we need to draw the diagram and then use the property of angles subtended by an arc in a circle. Also, Angle subtended by any straight line when moved from one point to another point forming an arc on the line is 180°.