Question

question 14 of 22 point d is the center of the given circle. what is the measure of $overparen{ac}$? a. $48^{circ}$ b. $84^{circ}$ c. $24^{circ}$ d. $96^{circ}$

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. Here, $\angle ABC = 48^{\circ}$ is an inscribed angle and $\angle ADC$ is the central angle subtending arc $\overset{\frown}{AC}$.

Step2: Calculate the central angle

If $\theta_{i}$ is the inscribed angle and $\theta_{c}$ is the central angle subtending the same arc, then $\theta_{c}=2\theta_{i}$. Given $\theta_{i} = 48^{\circ}$, so $\theta_{c}=2\times48^{\circ}$.
$2\times48^{\circ}=96^{\circ}$
The measure of an arc is equal to the measure of its central angle. So the measure of $\overset{\frown}{AC}$ is $96^{\circ}$.

Answer:

D. $96^{\circ}$