Question

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.

Hint:

The angle between two tangents drawn from an external point is supplementary to the angle subtended by the line-segment joining the points of contact at the center.

Answer 1

Step 1: Define the setup.
Let the center of the circle be \( O \), and let \( P \) be an external point from which two tangents \( PA \) and \( PB \) are drawn to the circle. The points of contact are \( A \) and \( B \). We are asked to prove that the angle \( \angle APB \) between the two tangents is supplementary to the angle \( \angle AOB \) subtended by the line-segment \( AB \) at the center.
Step 2: Use the property of tangents.
We know that the angle between two tangents drawn from an external point to a circle is half of the angle subtended by the line-segment joining the points of contact at the center. That is: \[ \angle APB = \frac{1}{2} \angle AOB \]
Step 3: Prove the supplementary angle.
Since the total angle around a point is \( 360^\circ \), the angle between the tangents and the angle subtended by the line-segment at the center must add up to \( 180^\circ \). Therefore, we have: \[ \angle APB + \angle AOB = 180^\circ \]
Step 4: Conclusion.
Thus, we have proved that the angle between the two tangents is supplementary to the angle subtended by the line-segment joining the points of contact at the center.