Question

which statement is true regarding the diagram of circle p?
○ the sum of y and z must be 2x.
○ the sum of y and z must be $\frac{1}{2}x$.
○ the difference of z and y must be 2x.
○ the difference of z and y must be $\frac{1}{2}x$.

Explanation:

Step1: Recall secant-tangent angle theorem

The measure of an angle formed by two secants (or a secant and tangent) outside a circle is half the difference of the measures of the intercepted arcs. For $\angle S$ (measure $x$), intercepted arcs are $z$ (major arc) and $y$ (minor arc).

Step2: Apply the theorem

$x = \frac{1}{2}(z - y)$

Step3: Rearrange the equation

Multiply both sides by 2: $2x = z - y$

Answer:

The difference of z and y must be 2x.