Question

question 21 of 25 point d is the center of the given circle. what is the measure of arc ac?

Explanation:

Step1: Recall central - inscribed angle relationship

The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.

Step2: Identify the inscribed and central angles

In the circle with center \(D\), \(\angle ABC = 48^{\circ}\) is an inscribed angle and \(\angle ADC\) is the central angle subtending arc \(\overset{\frown}{AC}\).

Step3: Calculate the measure of the central angle

Since the measure of an inscribed angle \(\theta_{i}\) and central angle \(\theta_{c}\) related by \(\theta_{i}=\frac{1}{2}\theta_{c}\), then \(\theta_{c} = 2\theta_{i}\). Given \(\theta_{i}=48^{\circ}\), so \(\theta_{c}=2\times48^{\circ}=96^{\circ}\). The measure of arc \(\overset{\frown}{AC}\) is equal to the measure of the central angle \(\angle ADC\), which is \(96^{\circ}\).

Answer:

D. 96°