Question

Circle Q is shown. Secant L N and tangent P N intersect at point N outside of the circle. Secant L N intersects the circle at point M. Arc M P is y, arc L P is x, and arc M L is z.
Which equation is correct regarding the measure of ∠MNP?

m∠MNP = One-half(x – y)
m∠MNP = One-half(x + y)
m∠MNP = One-half(z + y)
m∠MNP = One-half(z – y)

Answer 1

Answer: FIRST OPTION.

Step-by-step explanation:

By definition, the "Angle formed by a Tangent and a Secant" is:

$Angle\ formed\ by\ Tangent\ and\ Secant=\frac{1}{2}({Difference\ of\ Intercepted\ Arcs)$

In this case you can identify in the figure that the Intercepted Arcs are:

 $LP=x$ and $PM=y$

And the Angle formed by the tangent and the secant is:

$\angle MNP$

Therefore, the following equation can be used to calculate the measure of the angle ∠MNP:

$m\angle MNP=\frac{1}{2}(LP-PM)\\\\m\angle MNP=\frac{1}{2}(x-y)$

This matches with the First option.

Answer 2

Answer:

A.

Step-by-step explanation: