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 $overparen{xz}$?
a. $\frac{1}{2}(247 + x)=67$
b. $\frac{1}{2}(247 - x)=67$
c. $\frac{1}{2}(247 - 67)=x$
d. $\frac{1}{2}(247 + 67)=x$

Explanation:

Step1: Recall tangent - secant angle formula

The measure of an angle formed by two tangents to a circle is half the difference of the measures of the intercepted arcs.
The larger arc is \(247^{\circ}\) and the smaller arc is \(x^{\circ}\), and the angle formed by the two tangents is \(67^{\circ}\).
The formula for the measure of the angle \(\theta\) formed by two tangents is \(\theta=\frac{1}{2}(m_{1}-m_{2})\), where \(m_{1}\) is the measure of the larger - intercepted arc and \(m_{2}\) is the measure of the smaller - intercepted arc.

Step2: Substitute the values into the formula

Substituting \(\theta = 67^{\circ}\), \(m_{1}=247^{\circ}\), and \(m_{2}=x^{\circ}\) into the formula \(\theta=\frac{1}{2}(m_{1}-m_{2})\), we get \(67=\frac{1}{2}(247 - x)\).

Answer:

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