Why Radius Is Perpendicular to a Tangent

Quick answer: At a tangent's point of contact T, the radius OT is perpendicular to the tangent. The perpendicular from the centre is the shortest distance to that line; a closer point would force the line to cut the circle.

Worked example

A tangent PT touches a circle of radius 5 cm. If OP = 13 cm, find PT.

1. OT = 5 cm and ∠OTP = 90°.
2. OP is the hypotenuse of triangle OTP.
3. PT² = OP² − OT² = 169 − 25 = 144.
4. PT = 12 cm.

Check the result

The lengths 5, 12, 13 satisfy Pythagoras. P lies outside because OP > radius.

Common mistake to avoid

The right angle is at T, not at the external point P. Use OP as the hypotenuse.

Try it yourself

1. Radius is 3 cm and OP = 5 cm; find PT. 2. Can an external tangent segment have OP smaller than the radius?

Answers and reasoning

1. PT = √(25 − 9) = 4 cm. 2. No: P would lie inside the circle, so no tangent from P exists.

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