Circles and Constructions
Circles and Constructions tests tangent properties and straightedge-and-compass constructions. The exam may ask for a proof, a tangent length, or the steps and justification of a construction.
Core rules
- A tangent touches a circle at exactly one point.
- The tangent at any point is perpendicular to the radius through that point.
- Tangents drawn from an external point to a circle are equal in length.
- The angle between two tangents from an external point is supplementary to the angle between the radii to the points of contact.
- Construction tasks include dividing a segment in a given ratio, constructing a triangle similar to a given triangle, and drawing tangents from an external point.
Worked example
From an external point P, a tangent PQ touches a circle of radius 5 cm. If OP=13 cm, find PQ. OQ is a radius to the point of contact, so OQ⊥PQ and triangle OQP is right-angled at Q. PQ=√(OP²-OQ²)=√(13²-5²)=√(169-25)=√144=12 cm.
Traps
Do not treat a tangent as a chord; it meets the circle at one point only. In tangent-length problems, the right angle is at the point of contact, not at the external point. Constructions must show arcs and justify why the figure satisfies the required condition. Do not erase construction arcs if the marking scheme requires them. Equal tangents are from the same external point only.