Question

In the given figure, PA is a tangent from an external point P to a circle with centre O. If ∠POB = 115°, then ∠APO is equal to ______.

Options

• 25°

• 65°

• 90°

• 35°

Answer 1

In the given figure, PA is a tangent from an external point P to a circle with centre O. If ∠POB = 115°, then ∠APO is equal to 25°.

Explanation:

Given: 

• PA is a tangent to the circle from external point P.
• ∠POB = 115°.
• We need to find ∠APO.

The line segment OB is the radius of the circle. The tangent PA is perpendicular to the radius OA at the point of contact A, so:

∠OAP = 90°

Since POB is a straight angle, the angle subtended by the arc AB at the center is:

∠AOB = 180° − ∠POB 

∠AOB = 180° − 115° = 65°

Since △APO is a right-angled triangle at A, using the angle sum property: 

∠APO = 90° − ∠AOB

∠APO = 90° − 65° = 25°