Line is tangent to a circle with center O at point T. Point P lies on line , PT = 24, and OP = 26. What is the radius of the circle?

Line is tangent to a circle with center O at point T. Point P...

Math · Geometry and TrigonometryMedium

Line $\ell$ is tangent to a circle with center $O$ at point $T$ . Point $P$ lies on line $\ell$ , $PT = 24$ , and $OP = 26$ . What is the radius of the circle?

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Answer: 10

A tangent line is perpendicular to the radius drawn to the point of tangency, so triangle $OTP$ has a right angle at $T$ and hypotenuse $\overline{OP}$ . By the Pythagorean theorem, $OT^2 + 24^2 = 26^2$ , so $OT^2 = 676 - 576 = 100$ and $OT = 10$ .

Treating $\overline{OP}$ as a leg instead of the hypotenuse gives $\sqrt{24^2 + 26^2} \approx 35.4$ , but the side opposite the right angle at $T$ is $\overline{OP}$ .

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