Circle Geometry: Chords and Angles

In short: Circle geometry is a set of rules about chords and angles in a circle. The angle at the centre is twice the angle at the circumference on the same arc, the angle in a semicircle is $90^{\circ }$, and opposite angles of a cyclic quadrilateral add to $180^{\circ }$.

Starting point: Years 9 and 10

Circle words

• Radius: centre to the edge. Diameter: right across through the centre.
• Chord: a straight line joining two points on the circle.
• Arc: part of the circumference. A chord splits the circle into a minor and a major segment.
• Subtend: a chord or arc "makes" an angle at a point. Angle $APB$ is subtended by chord $AB$.
• A cyclic quadrilateral has all four corners on the circle.

Chord properties

• A line from the centre perpendicular to a chord bisects it.
• Chords of equal length are the same distance from the centre.
• Equal chords subtend equal angles at the centre.

Angle theorems

Theorem 3 is theorem 1 with a straight angle: $180^{\circ }$ at the centre gives $90^{\circ }$ at the edge.

Quick summary

• A perpendicular from the centre bisects a chord. Use Pythagoras with half the chord.
• The angle at the centre is twice the angle at the circumference on the same arc.
• Angles in the same segment are equal. The angle in a semicircle is $90^{\circ }$.
• Opposite angles of a cyclic quadrilateral add to $180^{\circ }$.
• Two radii form an isosceles triangle.

Quiz