Tangents: Radius, Equal Tangents and Alternate Segment
What the exam tests
Tangents questions test the tangent–radius perpendicularity, equal tangent lengths from an external point, the angle between two tangents, and the alternate segment theorem. Many problems reduce to a right triangle formed by the centre, the external point and the point of contact.
Core rules
- A tangent is perpendicular to the radius at the point of contact.
- Tangents drawn from the same external point are equal in length.
- If PA and PB are tangents from P, then PA=PB and ∠APB+∠AOB=180°, where O is the centre.
- For an external point P at distance OP from the centre, tangent length =√(OP²−r²).
- Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
Worked example
A circle of radius 5 cm has an external point P with OP=13 cm. Let PA be a tangent, with A the point of contact. OA is perpendicular to PA, so triangle OAP is right-angled at A. Then PA²=OP²−OA²=13²−5²=169−25=144, so PA=12 cm. The other tangent from P has the same length.
Traps
Do not use OP−r as the tangent length; use the right-triangle relation. The radius meets the tangent at the point of contact, not at the external point. Equal tangents come from the same external point only. In the alternate segment theorem, the angle is between the tangent and the chord at the point of contact. Check whether the question gives a diameter or a radius.