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

A.

Explanation:

Answer 2

Answer: FIRST OPTION.

Explanation:

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

`Angle\ formed\ by\ Tangent\ and\ Secant=(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=(1)/(2)(LP-PM)\\\\m\angle MNP=(1)/(2)(x-y)`

This matches with the First option.