Question

Line segment KL is tangent to circle J at point K.

Circle J is shown. Line segment J K is a radius with length r. Line segment K L is a tangent with length 24 and it intersects the circle at point K. A line is drawn from point L to point J and goes through a point on the circle. The length from point L to the point on the circle is 16. The length from the point on the circle to center point J is r.

What is the length of the radius, r?

8 units
10 units
12 units
16 units

Answer 1

Let the point where LJ meets the circle be P. Then LP = 16 and PJ = r, so LJ = LP + PJ = 16 + r.

Because KL is tangent and JK is a radius, JK ⟂ KL, so triangle JKL is right with legs r and 24. Hence LJ = sqrt(r^2 + 24^2) = sqrt(r^2 + 576).

Set equal: 16 + r = sqrt(r^2 + 576). Square both sides:
(r + 16)^2 = r^2 + 576 → r^2 + 32r + 256 = r^2 + 576 → 32r = 320 → r = 10.

Answer: 10 units.