Question

QR is tangent to circle P at point Q.

Circle P is shown. Line segment P Q is a radius. Line segment Q R is a tangent that intersects the circle at point Q. A line is drawn from point R to point P and goes through a point on the circle. The length of R Q is 5.3 and the length of Q P is 3.

What is the approximate length of RP? Round to the nearest tenth.

5.6 units
6.1 units
8.3 units
9.8 units

Answer 1

Use the tangent–secant theorem: (tangent length)^2 = (external secant segment) × (whole secant).
Let RP = x. The secant through the center meets the near and far circle points at distances x − 3 and x + 3 from R, so

(5.3)^2 = (x − 3)(x + 3) = x^2 − 9.
28.09 = x^2 − 9 → x^2 = 37.09 → x ≈ √37.09 ≈ 6.0908 ≈ 6.1.

Answer: 6.1 units.