Question

AB is the longest chord of a circle with centre O. P is any point on the circumference of the circle, not coinciding with A or B. Find $\angle APB$ in degrees.

• A. 90${}^{\circ }$
• B. 100${}^{\circ }$
• C. 80${}^{\circ }$
• D. 75${}^{\circ }$

Answer 1

The correct option is A 90${}^{\circ }$


Consider the circle as shown, with centre O and AB as the longest chord. Since diameter is the longest chord, AB is the diameter.
Recall that the angle subtended by an arc at the centre of the circle is twice the angle subtended by this arc in the other segment.
Now the angle subtended by the arc at the centre is $\angle AOB$, which is ${180}^{\circ }$. Let P be any point on the remaining part of the circle. Different positions of P are shown. In each case, $\angle APB$ is the angle subtended by the arc in the other segment. So it should be half the angle subtended by the arc at the centre (which is ${180}^{\circ }$).
Half of ${180}^{\circ }$ is ${90}^{\circ }$. So $\angle APB={90}^{\circ }$.