Circles

The One Thing

Circle geometry is governed by equal distances from the centre and the symmetry of chords, arcs, radii, and diameters. The central result is that the angle formed at the centre by an arc is twice the angle formed by the same arc at any point on the remaining circle.

Definitions and Results

• Circle: A closed plane curve whose every point is at the same distance from a fixed point.
• Centre: The fixed point inside a circle from which every point on the circle is equally distant.
• Radius: A line segment joining the centre to any point on the circle.
• Diameter: A chord passing through the centre; it is the longest chord of a circle.
• Chord: A line segment joining any two points on a circle.
• Arc: A part of the circumference of a circle between two points.
• Circumference: The complete boundary or perimeter of a circle.
• Semicircle: Half of a circle formed by a diameter and one of the two corresponding arcs.
• Sector: The region enclosed by two radii and the arc between their endpoints.
• Segment: The region enclosed by a chord and the arc joining the endpoints of that chord.
• Central angle: An angle whose vertex is at the centre of the circle.
• Inscribed angle: An angle whose vertex lies on the circle and whose arms are chords.
• Cyclic quadrilateral: A quadrilateral whose all four vertices lie on the same circle.
• Concyclic points: Points that lie on one common circle.
• Equal chords: Chords of the same circle having equal lengths.
• Perpendicular distance from centre: The shortest distance from the centre to a chord, measured along a perpendicular.
• Radius and diameter: All radii of the same circle are equal. The diameter is twice the radius:

$$d=2r,r=\frac{d}{2}.$$

• Circumference: The circumference of a circle is

$$2\pi r\text{or}\pi d.$$ Use $\pi$ approximately as $22/7$ or $3.14$ when the problem specifies or requires an approximation.

• Diameter and semicircles: A diameter divides a circle into two equal semicircles and is the longest chord of the circle.
• Equal chords and central angles: Equal chords of the same circle subtend equal angles at the centre.
• Equal chords and distance from the centre: Equal chords of the same circle are equidistant from the centre, and chords equidistant from the centre are equal.
• Perpendicular from the centre: The perpendicular from the centre of a circle to a chord bisects the chord.
• Midpoint of a chord: The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
• Central and inscribed angles: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining part of the circle.
• Angles in the same segment: Angles in the same segment of a circle are equal when they stand on the same chord or arc.
• Angle in a semicircle: The angle in a semicircle is a right angle. Therefore, an angle subtended by a diameter is $90^{\circ }$.
• Cyclic quadrilateral: Opposite angles of a cyclic quadrilateral are supplementary:

$$\text{opposite angles sum to }180^{\circ }.$$ An exterior angle of a cyclic quadrilateral equals the opposite interior angle.

• Determining a circle: A circle can be uniquely determined by three non-collinear points.
• Perpendicular bisectors: The perpendicular bisectors of two chords meet at the centre of the circle.

Worked Methods

Proving results using equal radii

• Join the centre to relevant points on the circle to form triangles.
• Use the fact that all radii of the same circle are equal.
• Apply side-side-side reasoning to prove the triangles congruent.
• Deduce equal corresponding angles or lengths.

This method establishes the connection between equal chords, equal central angles, and equal distances from the centre.

Showing that a perpendicular from the centre bisects a chord

• Let $AB$ be a chord and let $O$ be the centre.
• Draw $OM$ perpendicular to $AB$, meeting the chord at $M$.
• Compare triangles $OMA$ and $OMB$.
• The radii $OA$ and $OB$ are equal.
• $OM$ is common to both triangles.
• Both triangles contain right angles at $M$.
• The triangles are congruent, so $AM=MB$.

Therefore, the perpendicular from the centre to a chord bisects the chord.

Showing that the line to the midpoint of a chord is perpendicular

• Let $AB$ be a chord and let $M$ be its midpoint, so $AM=MB$.
• Join the centre $O$ to $A$, $B$, and $M$.
• Compare triangles $OMA$ and $OMB$.
• The radii $OA$ and $OB$ are equal.
• $AM=MB$ by the definition of midpoint.
• $OM$ is common.
• The triangles are congruent by side-side-side reasoning.
• The angles at $M$ are equal and form a straight line, so each is $90^{\circ }$.

Thus, $OM\perp AB$.

Using equal chords

• Identify the equal chords in the same circle.
• Use the theorem that equal chords subtend equal central angles.
• Alternatively, use the theorem that equal chords are equidistant from the centre.
• If chords are equidistant from the centre, apply the converse theorem to conclude that they are equal.

Using the central-angle and inscribed-angle theorem

• Identify the arc or chord on which the angle stands.
• Identify the corresponding central angle and inscribed angle.
• Apply

$$\text{central angle}=2\times \text{inscribed angle}.$$

• Rearrange if necessary:

$$\text{inscribed angle}=\frac{1}{2}\text{central angle}.$$

This applies when the two angles subtend the same arc and the inscribed angle has its vertex on the remaining part of the circle.

Using angles in the same segment

• Identify two angles whose vertices lie on the same segment of the circle.
• Check that both angles stand on the same chord or arc.
• Set the angles equal:

$$\angle A=\angle B.$$

Using the angle in a semicircle

• Identify whether the angle is subtended by a diameter.
• The diameter forms a semicircle, whose central angle is $180^{\circ }$.
• Use the central-angle and inscribed-angle theorem:

$$\text{inscribed angle}=\frac{1}{2}(180^{\circ })=90^{\circ }.$$

Using a cyclic quadrilateral

• Confirm that all four vertices lie on the same circle.
• Identify a pair of opposite interior angles.
• Apply

$$\angle A+\angle C=180^{\circ },\angle B+\angle D=180^{\circ }.$$

• If an exterior angle is given, equate it to the opposite interior angle.

Determining the centre from chords

• Identify two chords of the circle.
• Construct the perpendicular bisector of each chord.
• Their point of intersection is the centre of the circle.
• This follows because the perpendicular bisectors of two chords meet at the centre.

Determining a circle from points

• Identify three non-collinear points.
• Construct the perpendicular bisectors of two segments joining pairs of the points.
• Their intersection gives the centre.
• The distance from this centre to any of the three points gives the radius.
• The circle through all three points is uniquely determined.

Where It Goes Wrong

• Forgetting that the equal-chord, equal-angle, and equal-distance results apply to chords of the same circle.
• Treating a diameter merely as a long chord and overlooking that it is the longest chord, divides the circle into two equal semicircles, and subtends a $90^{\circ }$ angle.
• Reversing the central-angle relationship: the central angle is twice the corresponding inscribed angle, not half of it.
• Applying “angles in the same segment are equal” without checking that the angles stand on the same chord or arc and lie in the relevant segment.
• Using the cyclic-quadrilateral result without confirming that all four vertices are concyclic; opposite angles are supplementary only when the quadrilateral is cyclic.
• Forgetting the required approximation for $\pi$, which may be $22/7$ or $3.14$ when specified.

What Gets Asked

• Define a circle, centre, radius, diameter, chord, arc, circumference, semicircle, sector, segment, central angle, inscribed angle, cyclic quadrilateral, or concyclic points.
• Calculate a radius, diameter, or circumference using $d=2r$, $r=d/2$, $2\pi r$, or $\pi d$.
• Prove that a perpendicular from the centre bisects a chord.
• Prove that the line from the centre to the midpoint of a chord is perpendicular to the chord.
• Use congruent triangles formed by equal radii to establish equal lengths or angles.
• Determine unknown angles using equal chords, the central-angle and inscribed-angle theorem, or angles in the same segment.
• Show that an angle subtended by a diameter is $90^{\circ }$.
• Find an unknown angle in a cyclic quadrilateral using supplementary opposite angles or the exterior-angle theorem.
• Determine the centre using perpendicular bisectors of two chords.
• Explain why three non-collinear points uniquely determine a circle.