Question

A circle is shown. Secant X Z and tangent W Z intersect at point Z outside of the circle. Secant X Z goes through the center of the circle and intersects the circle at point Y. The length of W Z is k + 4, the length of Z Y is k, and the length of X Y is 12.
What is the length of line segment XZ?

4 units
8 units
16 units
20 units

Answer 1

Use the tangent-secant power theorem: (tangent length)^2 = (external secant segment)·(entire secant).

Here ZW = k+4, ZY = k, and ZX = ZY + YX = k + 12. So
(k+4)^2 = k(k+12).
Expand: k^2 + 8k + 16 = k^2 + 12k → 16 = 4k → k = 4.
Then XZ = k + 12 = 4 + 12 = 16.

Answer: 16 units.