Question

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

Explanation:

Step1: Recall the tangent-secant angle theorem

The measure of an angle formed by a tangent and a secant (or chord) intersecting outside a circle is half the difference of the measures of the intercepted arcs. The formula is \( m\angle = \frac{1}{2}(\text{measure of the larger arc} - \text{measure of the smaller arc}) \).

Step2: Identify the intercepted arcs

In the diagram, the tangent is \( NP \) and the secant is \( NML \). The larger intercepted arc is \( z \) (arc \( L M \)) and the smaller intercepted arc is \( y \) (arc \( M P \)). So the angle \( \angle MNP \) is formed by the tangent \( NP \) and the secant \( NM \), so by the theorem:
\( m\angle MNP=\frac{1}{2}(z - y) \)

Answer:

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