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 a circle, chord \( AB \) forms a \( 180^\circ \) angle at the center. Since \( \angle DCO \) involves the same chord, the central angle is twice the inscribed angle. Therefore, \[\nx = \frac{180^\circ - 120^\circ}{2} = 30^\circ\n\] Using the angle sum property in triangle \( ABC \), we calculate: \[\ny = 180^\circ - 130^\circ - x = 180^\circ - 130^\circ - 30^\circ = 60^\circ\n\] The solution is \( x = 50^\circ, y = 60^\circ \).