Question

which equation is correct regarding the measure of ∠mnp?
○ ( m angle mnp=\frac{1}{2}(x - y) )
○ ( m angle mnp=\frac{1}{2}(x + y) )
○ ( m angle mnp=\frac{1}{2}(z + y) )
○ ( m angle mnp=\frac{1}{2}(z - y) )

Explanation:

Step1: Recall the formula for the measure of an angle formed by a secant and a tangent

The measure of an angle formed by a secant and a tangent outside a circle is half the difference of the measures of the intercepted arcs.

Step2: Identify the intercepted arcs

In the given circle, the angle \( \angle MNP\) is formed by a secant \(NL\) and a tangent \(NP\). The intercepted arcs are arc \(z\) and arc \(y\).

Step3: Apply the formula

By the formula \(m\angle=\frac{1}{2}(\text{larger arc}-\text{smaller arc})\), so \(m\angle MNP=\frac{1}{2}(z - y)\)

Answer:

\(m\angle MNP=\frac{1}{2}(z - y)\) (the fourth option)