Question

in the diagram below, $overline{xy}$ and $overline{yz}$ are tangent to $odot o$. which equation could be solved to find $x$, the measure of $overline{yz}$?
a. $\frac{1}{2}(247 - 67)=x$
b. $\frac{1}{2}(247 + 67)=x$
c. $\frac{1}{2}(247 - x)=67$
d. $\frac{1}{2}(247 + x)=67$

Explanation:

Step1: Recall tangent - arc 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 intercepted arcs

Let the major arc be $247^{\circ}$ and the minor arc be $x$. The angle formed by the two tangents is $67^{\circ}$.

Step3: Apply the formula

The formula for the measure of the angle formed by two tangents is $\theta=\frac{1}{2}(m_{major\ arc}-m_{minor\ arc})$. Substituting $\theta = 67^{\circ}$, $m_{major\ arc}=247^{\circ}$ and $m_{minor\ arc}=x$, we get $\frac{1}{2}(247 - x)=67$.

Answer:

C. $\frac{1}{2}(247 - x)=67$