Circles
Circle questions test tangent theorems and the standard configurations: two tangents from an external point, a tangent with a chord, and a quadrilateral formed by two radii and two tangents. The theorems are short, so the marks are in the justification.
Core rules
- A tangent is perpendicular to the radius at the point of contact.
- From an external point, the two tangents are equal in length, and the line from the external point to the centre bisects the angle between them.
- In the configuration with external point P, centre O and contact points A, B: ∠APB + ∠AOB = 180°, because OAPB is cyclic with right angles at A and B.
- From a point on the circle exactly one tangent can be drawn; from an external point two; from an interior point none.
- The perpendicular from the centre to a chord bisects the chord, and equal chords are equidistant from the centre.
- Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
Worked example
From P, with OP = 13 cm and radius 5 cm, tangents PA and PB touch the circle at A and B. Since OA ⟂ PA, triangle OAP is right-angled: PA = √(13² − 5²) = √144 = 12 cm, and PB = 12 cm too. Area of quadrilateral OAPB = 2 × ½ × 5 × 12 = 60 cm².
Traps
- Assuming the tangent length equals OP or the radius.
- Using the diameter where the radius belongs in Pythagoras.
- Stating the tangent–radius perpendicularity without naming the point of contact.
- Claiming a tangent meets the circle at two points.
- Invoking the alternate segment theorem with the wrong chord.