Question

point o is the center of the given circle. what is the measure of (widehat{yz})?
image of a circle with center o, points x, y, z on the circumference; triangle xoy has a 54° angle at x
a. (54^circ)
b. (27^circ)
c. (72^circ)
d. (108^circ)

Explanation:

Step1: Identify triangle type

Triangle \( OXY \) is isosceles (\( OX = OY \), radii), so \( \angle OXY=\angle OYX = 54^\circ \).

Step2: Find \( \angle XOY \)

Sum of angles in triangle: \( \angle XOY=180^\circ - 54^\circ - 54^\circ = 72^\circ \).

Step3: Analyze straight line \( XZ \)

\( XZ \) is diameter, so \( \angle XOY+\angle YOZ = 180^\circ \).

Step4: Calculate \( \angle YOZ \) (arc \( YZ \) measure)

\( \angle YOZ = 180^\circ - 72^\circ = 108^\circ \)? Wait, no—wait, arc \( YZ \) is central angle \( \angle YOZ \)? Wait, no, wait: Wait, \( OX \) and \( OZ \) are diameter, so \( \angle XOZ = 180^\circ \). Wait, maybe I messed up. Wait, \( \angle OXY = 54^\circ \), \( OX = OY \), so \( \angle OXY = \angle OYX = 54^\circ \), so \( \angle XOY = 180 - 2*54 = 72^\circ \). Then, since \( XZ \) is a straight line (diameter), \( \angle YOZ = 180 - \angle XOY = 180 - 72 = 108^\circ \)? But wait, no—wait, the arc \( YZ \) is the central angle \( \angle YOZ \). Wait, but let's check again. Wait, maybe the triangle is \( OXY \), with \( OX = OY \) (radii), so base angles equal. Then \( \angle XOY = 180 - 2*54 = 72 \). Then, since \( XZ \) is a diameter, \( \angle YOZ = 180 - 72 = 108 \). So arc \( YZ \) is \( 108^\circ \). Wait, but let's confirm: The central angle for arc \( YZ \) is \( \angle YOZ \). Since \( XZ \) is a straight line (180 degrees), and \( \angle XOY = 72^\circ \), then \( \angle YOZ = 180 - 72 = 108^\circ \). So the measure of arc \( YZ \) is \( 108^\circ \).

Wait, but let's check the options. Option D is \( 108^\circ \). So that's the answer.

Answer:

D. \( 108^\circ \)