Question

in the diagram below, $overline{xy}$ and $overline{yz}$ are tangent to $odot o$. what is the measure of $overparen{xwz}$? a. $180^{circ}$ b. $224^{circ}$ c. $88^{circ}$ d. $248^{circ}$

Explanation:

Step1: Recall tangent - radius property

The radius is perpendicular to the tangent at the point of tangency. So, $\angle OXY = 90^{\circ}$ and $\angle OZY=90^{\circ}$.

Step2: Find the measure of $\angle XOZ$

In quadrilateral $OXYZ$, the sum of interior angles is $360^{\circ}$. Given $\angle XYZ = 68^{\circ}$, $\angle OXY = 90^{\circ}$, $\angle OZY = 90^{\circ}$. Let $\angle XOZ=x$. Then $90^{\circ}+90^{\circ}+68^{\circ}+x = 360^{\circ}$, so $x=360^{\circ}-(90^{\circ}+90^{\circ}+68^{\circ})=112^{\circ}$.

Step3: Find the measure of arc $XWZ$

The measure of an arc is related to the central - angle that subtends it. The measure of the major arc $XWZ$ and the central - angle $\angle XOZ$ are related as the measure of the major arc is $360^{\circ}-\text{measure of the minor arc}$. The minor arc $XZ$ has a central - angle measure of $112^{\circ}$. So the measure of arc $XWZ=360^{\circ}-112^{\circ}=224^{\circ}$.

Answer:

B. 224°