Question

In the given figure, PA and PB are tangents to a circle centred at O. If \( \angle AOB = 130^\circ \), then \( \angle APB \) is equal to :

• A. \( 130^\circ \)
• B. \( 50^\circ \)
• C. \( 120^\circ \)
• D. \( 90^\circ \)

Hint:

Just subtract the central angle from 180 to find the angle between tangents.

Answer 1

The Correct Option is B

To find \( \angle APB \) in the given diagram where PA and PB are tangents to the circle at points A and B respectively, and the circle is centered at O with \( \angle AOB = 130^\circ \), we can apply the following theorem about tangents and circles:

When two tangents are drawn to a circle from an external point, the angle between the tangents is equal to the difference between 180 degrees and the angle subtended by the line segments joining the points of tangency to the center of the circle.

Given:

• \( \angle AOB = 130^\circ \)

According to the property stated above, we can calculate \( \angle APB \) as follows:

\[ \angle APB = 180^\circ - \angle AOB \]

Substitute the known value:

\[ \angle APB = 180^\circ - 130^\circ = 50^\circ \]

Therefore, the angle \( \angle APB \) is \(50^\circ\).

Option Explanation:

• \( 130^\circ \): Incorrect, as it ignores the tangent property.
• \( 50^\circ \): Correct, calculated using the tangent property.
• \( 120^\circ \): Incorrect, irrelevant to the calculation.
• \( 90^\circ \): Incorrect, unrelated to the problem's conditions.