Question

LK is tangent to circle J at point K.

Circle J is shown. Line segment J K is a radius. Line segment K L is a tangent that intersects at point K. A line is drawn from point L through a point on the circle to the center point J. The length of the radius is r, the length of K L is 11, and the length of the line segment from point L to the point on the circle is 6.

What is the length of the radius?

StartFraction 6 Over 85 EndFraction
StartFraction 85 Over 12 EndFraction
StartFraction 121 Over 36 EndFraction
StartFraction 157 Over 12 EndFraction

Answer 1

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

The secant through the center is a diameter, so the far intersection is 2r beyond the near one. With external part 6, whole secant = 6 + 2r. Thus
121 = 6(6 + 2r) = 36 + 12r,
so 12r = 85 and r = 85/12.

Answer: 85/12.