Question

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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 theorem: (tangent)^2 = (external secant segment) × (whole secant).

External segment = NP = 11.5. Whole secant = NP + MN = 11.5 + 24 = 35.5.

n^2 = 11.5 × 35.5 = 408.25, so n = √408.25 ≈ 20.21, which rounds to 20.

Answer: 20 units.