Question

Circle X is shown. Line segment X Y is a radius. Line segment Y Z is a tangent that intersects the circle at point Y. A line is drawn from point Z to point X and goes through a point on the circle. The length of the line segment from point X to the point on the circle is 8, and the length of the line segment from the point on the circle to point Z is 9.
What must be the length of ZY in order for ZY to be tangent to circle X at point Y?

14 units
15 units
16 units
17 units

Answer 1

15 units.

Reason: XP = radius = 8, so the secant through the center meets the circle at two opposite points (distance 2r = 16 apart). The external secant segment is 9, so the whole secant from Z to the far intersection is 9 + 16 = 25. By the tangent-secant theorem, ZY^2 = (external secant)(whole secant) = 9·25 = 225, so ZY = 15.