Circles — Study Notes

1. Introduction

The circle is perhaps the most perfect figure in geometry. A circle is the collection of all points in a plane that are at a fixed distance from a fixed point. The fixed point is called the centre and the fixed distance is called the radius. Wheels, coins, clock faces, the cross-section of a pipe and the orbits of planets all remind us of circles. Because a circle has no corners and is perfectly symmetric, it has a rich set of beautiful properties.

In this chapter we will study the basic terms associated with a circle: the radius, diameter, chord, arc, segment and sector. We will then prove several important theorems: equal chords subtend equal angles at the centre and are equidistant from the centre, the perpendicular from the centre to a chord bisects the chord, and the angle subtended by an arc at the centre is twice the angle subtended at any point on the remaining part of the circle. We will also study cyclic quadrilaterals and the property that their opposite angles are supplementary.

2. Basic Terms Related to a Circle

• Centre: The fixed point O from which all points of the circle are equidistant.
• Radius: The distance from the centre to any point on the circle, written r. All radii of a circle are equal.
• Diameter: The longest chord of the circle, passing through the centre. The diameter equals twice the radius: d = 2r.
• Chord: A line segment whose endpoints lie on the circle.
• Arc: A part of the circle between two points. A minor arc is smaller than a semicircle and a major arc is larger.
• Segment: The region bounded by a chord and the arc it subtends.
• Sector: The region bounded by two radii and the arc between them.
• Circumference: The perimeter of the circle, equal to 2 pi r, where pi is approximately 22/7 or 3.14.

Important Facts

• Every circle is a closed figure in a plane.
• A chord divides a circle into a minor segment and a major segment.
• The longest chord is the diameter, so the diameter is twice the radius.
• Concentric circles have the same centre but different radii.

3. Equal Chords and Their Properties

Theorem 1: Equal Chords Subtend Equal Angles at the Centre

If two chords of a circle are equal, then they subtend equal angles at the centre. Conversely, chords that subtend equal angles at the centre are equal.

Proof: Let AB and CD be equal chords of a circle with centre O. Join OA, OB, OC and OD. In triangles AOB and COD, we have OA = OC and OB = OD (radii) and AB = CD (given). Hence the triangles are congruent by SSS, so angle AOB = angle COD.

Theorem 2: The Perpendicular from the Centre to a Chord Bisects the Chord

The perpendicular drawn from the centre of a circle to a chord bisects the chord. Conversely, the line joining the centre to the midpoint of a chord is perpendicular to the chord.

Theorem 3: Equal Chords are Equidistant from the Centre

Equal chords of a circle are equidistant from the centre. Conversely, chords that are equidistant from the centre are equal.

In other words, if chords AB and CD are equal, then the perpendicular distances from the centre O to the two chords are equal. And if two chords are at the same distance from the centre, then the chords are equal.

4. Angles in a Circle

Theorem 4: Angle at the Centre is Twice the Angle at the Circumference

The angle subtended by an arc at the centre of a circle is twice the angle subtended by it at any point on the remaining part of the circle.

If arc AB subtends angle AOB at the centre and angle ACB at a point C on the remaining part, then angle AOB = 2 x angle ACB.

Corollaries

• The angle in a semicircle is a right angle. Since the diameter subtends an angle of 180 degrees at the centre, any angle in the semicircle is 180/2 = 90 degrees.
• Angles in the same segment of a circle are equal. If two points C and D lie on the same arc, then angle ACB = angle ADB.
• An arc subtends equal angles at all points on the remaining part of the circle.

5. Cyclic Quadrilaterals

A quadrilateral is called a cyclic quadrilateral if all four of its vertices lie on a circle.

Theorem 5: Opposite Angles of a Cyclic Quadrilateral

The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.

If ABCD is a cyclic quadrilateral, then angle A + angle C = 180 degrees and angle B + angle D = 180 degrees. This is a very important property used to solve many circle problems. The converse is also true: if the sum of a pair of opposite angles of a quadrilateral is 180 degrees, then the quadrilateral is cyclic.

Theorems on More than One Chord

If two chords AB and CD of a circle intersect inside the circle at a point P, then the angle between them satisfies: angle APC = (1/2) x (sum of angles subtended by arcs AC and BD at the centre). Such intersecting chord results link arcs and angles in a unified way.

Quick Revision Tables

Table 1: Basic Terms of a Circle

Term	Definition
Radius	Distance from centre to a point on the circle
Diameter	Longest chord, equal to 2 x radius
Chord	Line segment joining two points on the circle
Arc	Part of the circle between two points
Segment	Region between a chord and its arc
Sector	Region between two radii and the arc
Circumference	2 pi r

Table 2: Important Theorems on Circles

Theorem	Statement
Equal chords	Subtend equal angles at the centre and are equidistant from the centre
Perpendicular from centre	Bisects the chord; converse also true
Angle at centre	Twice the angle at the circumference on the same arc
Angle in semicircle	Right angle (90 degrees)
Angles in same segment	Equal
Cyclic quadrilateral	Opposite angles sum to 180 degrees

Mind Map

graph TD
    A["Circles"] --> B["Basic terms"]
    B --> C["Centre, radius, diameter, chord"]
    B --> D["Arc, segment, sector, circumference"]
    A --> E["Equal chords"]
    E --> F["Equal angles at centre"]
    E --> G["Equidistant from centre"]
    A --> H["Perpendicular from centre to chord"]
    H --> I["Bisects the chord"]
    A --> J["Angles in a circle"]
    J --> K["Angle at centre = 2 x angle at circumference"]
    J --> L["Angle in semicircle = 90 degrees"]
    A --> M["Cyclic quadrilateral"]
    M --> N["Opposite angles sum to 180 degrees"]

Important Diagrams (SVG)

Diagram 1: Basic Parts of a Circle

Diagram 2: Angle at the Centre is Twice the Angle at the Circumference

Common Mistakes

1. Students often confuse a sector with a segment. A sector is bounded by two radii and an arc, while a segment is bounded by a chord and an arc.
2. A frequent error is stating that all chords are diameters. Only the chord passing through the centre is a diameter.
3. Students sometimes claim that the perpendicular from the centre bisects the angle at the centre; it bisects the chord, not the angle.
4. Many students forget that the angle in a semicircle is 90 degrees and instead write 180 degrees; 180 is the angle at the centre.
5. Students mix up the minor and major arcs, thinking the shorter arc is always the major arc; the shorter arc is the minor arc.
6. When applying the cyclic quadrilateral property, students wrongly add adjacent angles; it is the opposite angles that sum to 180 degrees.
7. Students forget the converse of the cyclic quadrilateral property and cannot identify that a quadrilateral is cyclic when opposite angles sum to 180 degrees.
8. A common error is using the angle subtended at the circumference instead of the centre when applying the double-angle theorem.

Exam Tips

1. In circle problems, always join the centre to the endpoints of the chord first; the perpendicular from the centre to the chord and the bisection property solve most questions.
2. Remember the phrase "angle in a semicircle is a right angle" and apply it immediately when you see a diameter.
3. For the double-angle theorem, first identify the arc, then the angle at the centre and the angle at the circumference on the same arc.
4. To prove a quadrilateral is cyclic, show that a pair of opposite angles sums to 180 degrees using the given angles.
5. Use the property that equal chords are equidistant from the centre to compare chords and their distances quickly.
6. Practise the standard theorem proofs, as "prove that the perpendicular from the centre of a circle to a chord bisects the chord" is frequently asked.
7. When two chords intersect, draw the figure carefully and use the intersecting chords results to relate the angles formed.

Conclusion

In this chapter we studied the basic terms of a circle: centre, radius, diameter, chord, arc, segment and sector. We proved that equal chords subtend equal angles at the centre and are equidistant from the centre, and that the perpendicular from the centre to a chord bisects it. We established the fundamental theorem that the angle subtended by an arc at the centre is twice the angle subtended at the circumference, leading to the results that the angle in a semicircle is a right angle and angles in the same segment are equal. Finally, we studied cyclic quadrilaterals and proved that their opposite angles are supplementary, with the converse used to recognise cyclic quadrilaterals. These circle theorems are powerful problem-solving tools and will be extended in higher classes to the study of tangents, sectors and the areas of circles.