Question

In the figure given below (not drawn to scale), A, B and C are three points on a circle with centre O. The chord BA is extended to a point T such that CT becomes a tangent to the circle at point C. If $\angle \angle ATC = 30^\circ$ and $\angle ACT = 50^\circ$, then the angle $\angle ABOA$ is:

• A. 100°
• B. 150°
• C. 80°
• D. Not possible to determine

Hint:

When working with angles in circles, use the properties of tangents and angles subtended by chords to solve for unknown angles.

Answer 1

The Correct Option is B

We are given that $BA$ is a chord of the circle and $CT$ is a tangent at point $C$. From the properties of tangents and circles, we know that the angle between the chord and the tangent is equal to the angle subtended by the chord at the opposite side of the circle. Hence, $\angle ACT = \angle ATC = 30^\circ$. Now, $\angle ATC = 30^\circ$ and $\angle ACT = 50^\circ$, so $\angle AOB$ can be found by using the sum of the angles in a circle. Thus, the Correct Answer is $150^\circ$.