Question

in the diagram below, $overline{xy}$ and $overline{yz}$ are tangent to $odot o$. what is the measure of $overparen{xz}$? a. $148^{circ}$ b. $90^{circ}$ c. $74^{circ}$ d. $106^{circ}$

Explanation:

Step1: Recall circle - angle relationship

The measure of an angle formed by two tangents to a circle is half the difference of the measures of the intercepted arcs.

Step2: Identify the arcs

The major arc is 254° and the minor arc $\overparen{XZ}$ is what we want to find. Let the measure of $\overparen{XZ}=x$. The sum of the major and minor arcs of a circle is 360°, so the other arc is 254° and $x + 254^{\circ}=360^{\circ}$, or we can use the formula for the angle formed by two tangents. The angle $\angle Y = 74^{\circ}$, and the formula for the angle formed by two tangents $\angle Y=\frac{1}{2}(\text{major arc}-\text{minor arc})$.

Step3: Apply the formula

We know that $\angle Y = 74^{\circ}$, and $\angle Y=\frac{1}{2}(254 - x)$. So, $74=\frac{1}{2}(254 - x)$. Multiply both sides by 2: $148 = 254 - x$. Then, solve for $x$: $x=254 - 148=148^{\circ}$.

Answer:

A. 148°