Circles and Constructions
Circles and Constructions tests tangent properties, number of tangents, and straightedge-and-compass constructions. A tangent touches the circle at exactly one point.
Core rules
- The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Tangents drawn from an external point to a circle are equal in length.
- From an external point, exactly two tangents can be drawn; from a point on the circle, one; from an interior point, none.
- The angle between two tangents from an external point is supplementary to the angle between the radii to the points of contact.
- Construction of tangents from an external point uses the midpoint of the segment joining the centre and the external point, then a circle with that midpoint as centre.
Worked example
A circle has radius 5 cm and an external point P is 13 cm from the centre O. Find the length of tangent PA. Since OA is perpendicular to PA, triangle OAP is right-angled at A. PA^2 = OP^2 - OA^2 = 13^2 - 5^2 = 169 - 25 = 144, so PA = 12 cm. The second tangent PB also has length 12 cm.
Traps
The tangent length is not OP. Do not use the radius as a side of the right triangle without noting the perpendicular at the point of contact. In constructions, keep the compass width unchanged when copying lengths and show necessary arcs. A line that meets a circle at two points is a secant, not a tangent.