Circles and Constructions
The exam tests tangent and chord properties, the equality of tangents from an external point, and standard ruler-and-compass constructions involving circles and similar triangles.
Core rules
- A tangent is perpendicular to the radius at the point of contact.
- Tangents drawn from an external point to a circle are equal in length.
- The perpendicular from the centre to a chord bisects the chord; equal chords are equidistant from the centre.
- The angle between two tangents from an external point is supplementary to the angle subtended by the radii at the centre.
- Construction of tangents from an external point P: draw the circle with OP as diameter; its intersections with the given circle are the tangent points.
Worked example
A point P is 13 cm from the centre O of a circle of radius 5 cm. Find the length of the tangent from P. The radius OT is perpendicular to tangent PT, so triangle OTP is right-angled at T. OT = 5 cm and OP = 13 cm. By Pythagoras, PT² = 13² − 5² = 169 − 25 = 144. PT = 12 cm.
Traps
- Using the diameter instead of the radius in Pythagoras.
- Drawing a tangent that is not perpendicular to the radius at the point of contact.
- Skipping construction arcs and labels that the marking scheme requires.
- Assuming a line is a tangent without the perpendicularity or equal-length condition.