Circles and Constructions
This section covers Circles and Constructions. The exam tests tangent properties, the number of tangents from a point, equal tangent lengths, and compass-and-ruler constructions.
Core rules
- A tangent touches a circle at exactly one point; a secant cuts it at two points.
- 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 there are exactly two tangents; from a point on the circle there is one; from an inside point there is none.
- To construct tangents from external point P to a circle with centre O: join OP, draw its perpendicular bisector to find midpoint M, draw a circle with centre M and radius MO, and join P to the two intersection points with the given circle.
Worked example
A point P is 10 cm from the centre O of a circle of radius 6 cm. Find the tangent length from P. The radius to the point of contact is perpendicular to the tangent, forming a right triangle with OP as hypotenuse. Tangent length = √(10² - 6²) = √64 = 8 cm.
Traps
Do not call a line a tangent if it crosses the circle twice. In construction, the perpendicular bisector must be drawn accurately; freehand placement loses marks. The radius used in the right triangle is the radius to the point of contact, not any other radius. Keep the compass width unchanged when marking equal arcs.