Question

what must be the length of $overline{zy}$ in order for $overline{zy}$ to be tangent to circle x at point y?○ 14 units○ 15 units○ 16 units○ 17 units

Explanation:

Step1: Identify total length XZ

$XZ = 8 + 9 = 17$

Step2: Recognize right triangle XYZ

If $\overline{ZY}$ is tangent, $\angle XYZ = 90^\circ$, so use Pythagorean theorem: $ZY^2 + XY^2 = XZ^2$

Step3: Solve for ZY

Substitute $XY=8$, $XZ=17$:
$$ZY = \sqrt{XZ^2 - XY^2} = \sqrt{17^2 - 8^2}$$
$$ZY = \sqrt{289 - 64} = \sqrt{225} = 15$$

Answer:

15 units