The language of circles
A circle is the set of all points in a plane at a fixed distance from a fixed point. The fixed point is the centre, and the fixed distance is the radius.
Term | Meaning |
|---|---|
Chord | line segment joining two points on the circle |
Diameter | chord passing through the centre; length = 2r |
Arc | part of the circumference between two points |
Sector | region between two radii and their arc |
Segment | region between a chord and its arc |
Secant | line meeting the circle at two points |
Tangent | line meeting the circle at one point |
The diameter is the longest chord. Every diameter divides the circle into two semicircles.
Chords and the centre
Several results connect a chord with the centre O.
Equal chords and equal central angles
In the same circle, or in congruent circles:
equal chords subtend equal angles at the centre,
chords that subtend equal central angles are equal.
To see why, join the endpoints of both chords to the centre. The radii are equal, and the included central angles are equal, so the triangles are congruent by SAS.
A perpendicular from the centre
The perpendicular from the centre of a circle to a chord bisects the chord.
If OM ⟂ AB, then AM = MB. Triangles OMA and OMB are congruent by RHS: OA = OB, OM is common, and both have a right angle.
The converse is also true: the line from the centre to the midpoint of a chord is perpendicular to the chord.
Distance from the centre
Equal chords are equidistant from the centre, and chords equidistant from the centre are equal. Of two unequal chords, the longer chord lies closer to the centre.
Example. A circle has radius 13 cm. A chord is 5 cm from the centre. The perpendicular from the centre bisects the chord. If half the chord is x, then x² + 5² = 13², so x² = 144 and x = 12. The whole chord is 24 cm.
Angles made by an arc
For an arc AB, the angle ∠AOB at the centre is twice the angle ∠ACB at any point C on the remaining part of the circle:
∠AOB = 2∠ACB
This gives two important results:
angles in the same segment are equal because they stand on the same chord,
the angle in a semicircle is 90° because the central angle made by a diameter is 180°.
Check which arc the angle stands on. Points on opposite sides of the same chord do not give equal angles; their angles are supplementary.
Cyclic quadrilaterals
A quadrilateral whose four vertices lie on a circle is a cyclic quadrilateral. Its opposite angles are supplementary:
∠A + ∠C = 180° and ∠B + ∠D = 180°.
The converse is useful too: if a pair of opposite angles of a quadrilateral adds to 180°, the quadrilateral is cyclic.
An exterior angle of a cyclic quadrilateral equals its interior opposite angle. This follows because the exterior angle and the adjacent interior angle add to 180°, as do the two opposite interior angles.
A proof pattern
Many circle proofs become short after adding radii:
Join the centre to useful points on the circle.
Mark all radii equal.
Look for congruent isosceles triangles.
Use the angle sum of a triangle or a cyclic quadrilateral.
Summary
Equal chords correspond to equal central angles and equal distances from the centre.
A perpendicular from the centre bisects a chord.
A central angle is twice an angle on the remaining circle standing on the same arc.
Angles in the same segment are equal; an angle in a semicircle is 90°.
Opposite angles of a cyclic quadrilateral add to 180°.