Circles and Tangents
The exam tests the tangent–radius perpendicularity, equal tangent lengths from an external point, the number of tangents from different positions, and angle relations involving two tangents.
Core rules
- A tangent touches the circle at exactly one point, the point of contact.
- The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Lengths of the two tangents drawn from an external point are equal.
- From a point inside the circle: no tangent; on the circle: one tangent; outside: two tangents.
- If PA and PB are tangents from P, then △OAP ≅ △OBP (RHS), so OP bisects ∠APB and ∠AOB.
- The angle between two tangents from an external point is 180° minus the central angle between the radii to the points of contact.
Worked example
PA and PB are tangents from P to a circle with centre O, and ∠APB = 60°. Since OA ⊥ PA and OB ⊥ PB, quadrilateral OAPB has angles 90°, 60°, 90° and ∠AOB. So ∠AOB = 360° − 240° = 120°. Also PA = PB = PB, and OP bisects ∠APB into 30°.
Traps
Drawing the tangent as if it passes through the centre; assuming the tangent length equals the radius; omitting the right angle at the point of contact; adding rather than subtracting in the 180° relation; confusing the angle between tangents with the central angle.