Question

Line segment QP is tangent to the circle.

A circle is shown. Secant M P and tangent Q P intersect at point P outside of the circle. Secant M P intersects the circle at point N. The length of Q P is n, the length of N P is 11.5, and the length of M N is 24.

What is the length of line segment QP? Round to the nearest unit.

13 units
17 units
18 units
20 units

Answer 1

Use the tangent–secant power theorem: (QP)^2 = (PN)(PM). Here PN = 11.5 and PM = PN + MN = 11.5 + 24 = 35.5, so

(QP)^2 = 11.5 × 35.5 = 408.25, hence QP = sqrt(408.25) ≈ 20.205 ≈ 20 (nearest unit).

Answer: 20 units.