Parts of a circle
A chord joins two points of the circle, a diameter is the longest chord, an arc is part of the circumference, and a tangent is a line that touches the circle at one point. A radius to the point of contact is perpendicular to the tangent.
Angle properties
| Property | Statement |
|---|---|
| Centre and circumference | The angle at the centre is twice the angle at the circumference on the same arc |
| Semicircle | The angle in a semicircle is 90° |
| Same segment | Angles in the same segment are equal |
| Cyclic quadrilateral | Opposite angles add up to 180° |
| Alternate segment | The angle between a tangent and a chord equals the angle in the alternate segment |
Chord properties
A line from the centre perpendicular to a chord bisects the chord. Equal chords are the same distance from the centre. If the radius is r and the perpendicular distance to a chord is d, the half-length of the chord is √(r² − d²).
Tangent properties
Two tangents drawn from the same outside point are equal in length. The length of a tangent from a point at a distance d from the centre is √(d² − r²). The angle between a tangent and a chord equals the angle in the alternate segment (the alternate segment theorem).
A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of a tangent from P.
- The radius to the point of contact is perpendicular to the tangent, giving a right triangle.
- Square root.
Answer: The tangent is 12 cm long.
Choose the radius and the distance from the centre.
Key terms
- Chord
- A line joining two points on the circle.
- Tangent
- A line touching the circle at one point.
- Cyclic quadrilateral
- A quadrilateral with all four vertices on a circle.
- Arc
- A part of the circumference.
Summary
- The angle at the centre is twice the angle at the circumference.
- The angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral add to 180°.
- A radius is perpendicular to a tangent at the point of contact.
- Tangents from one point are equal.