Question

what must be the length of zy in order for zy to be tangent to circle x at point y? 14 units 15 units 17 units 10 units

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So, triangle $XYZ$ is a right - triangle with right angle at $Y$, and $XY$ is the radius, $ZY$ is the tangent, and $XZ$ is the hypotenuse.

Step2: Identify the lengths of the sides

Given that the radius $XY = 8$ and $XZ=8 + 9=17$.

Step3: Apply the Pythagorean theorem

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

Step4: Calculate the length of $ZY$

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

Answer:

15 units