Question

In the given figure, PA is a tangent from an external point P to a circle with centre O. If \(\angle POB = 125^\circ\), then \(\angle APO\) is equal to :

• A. 25°
• B. 65°
• C. 90°
• D. 35°

Hint:

Alternatively, find \(\angle AOP = 180^\circ - 125^\circ = 55^\circ\) using linear pair, then use angle sum property in \(\triangle OAP\).

Answer 1

The Correct Option is D

To solve for \(\angle APO\), we need to understand the properties of the circle and the tangent line.

1. The given angle \(\angle POB = 125^\circ\) refers to the angle between the radius \(OB\) and a line from the external point \(P\) that intersects the circle at \(B\).
2. In circle geometry, the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. This is known as the Alternate Segment Theorem.
3. Thus, the angle \(\angle APO\) is equal to \(\angle OBA\).
4. Because the straight line through \(O\) and \(B\) is 180°, and \(\angle POB\) is 125°, the supplementary angle \(\angle OBA = 180° - 125° = 55°\).
5. However, we need the angle \(\angle APO\), which is complementary to \(\angle OBA\) in the right triangle \(\triangle APO\). Therefore, \(\angle APO = 90° - \angle OBA = 90° - 55° = 35°\).

Thus, the angle \(\angle APO\) is 35°.