what must be the length of $overline{zy}$ in order for $overline{zy}$ to be tangent to circle x at point…

Answer:

B. 15 units

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, $\angle{XYZ}=90^{\circ}$, and $\triangle{XYZ}$ is a right - triangle.

Step2: Identify the lengths of the sides of the right - triangle

The radius of circle $X$ is $XY = 8$ units and $XZ=8 + 9=17$ units.

Step3: Apply the Pythagorean theorem

In right - triangle $\triangle{XYZ}$, by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = XZ$, $a = XY$, and $b = ZY$. We want to find $ZY$. So, $ZY=\sqrt{XZ^{2}-XY^{2}}$.

Step4: Substitute the values

Substitute $XZ = 17$ and $XY = 8$ into the formula: $ZY=\sqrt{17^{2}-8^{2}}=\sqrt{(17 + 8)(17 - 8)}=\sqrt{25\times9}=\sqrt{225}=15$ units.