Question

Use the following figure to find \( x^\circ \) and \( y^\circ \):

• A. \( x = 50^\circ, y = 30^\circ \)
• B. \( x = 30^\circ, y = 50^\circ \)
• C. \( x = 50^\circ, y = 60^\circ \)
• D. \( x = 55^\circ, y = 65^\circ \)

Hint:

In a circle, the angle subtended by a chord at the center is twice the angle subtended at any point on the circumference. Use this property to solve angle-related problems.

Answer 1

The Correct Option is C

In the given circle, the angle subtended by the chord \( AB \) at the center of the circle is \( 180^\circ \). Since \( \angle DCO \) is subtended by the same chord \( AB \), we can use the property that the angle subtended at the center is twice the angle subtended at the circumference. Thus, \[ x = \frac{180^\circ - 120^\circ}{2} = 30^\circ \] Now, using the angle sum property in triangle \( ABC \), we can find: \[ y = 180^\circ - 130^\circ - x = 180^\circ - 130^\circ - 30^\circ = 60^\circ \] Thus, the correct answer is \( x = 50^\circ, y = 60^\circ \).