I’m Up and Down, and Round and Round · Lesson 8 of 8
Chapter Summary and Practice
“Round up your circle facts before the questions come full circle.”
• Recall the major definitions and circle theorems. • Choose appropriate chord and arc properties in mixed problems. • Use circumcentre and perpendicular-bisector ideas confidently. • Apply cyclic-quadrilateral properties. • Solve multi-step problems using Pythagoras, congruence and circle geometry.
This chapter developed the geometry of circles from one simple defining idea: every point on a circle is the same distance from its centre. From that fact came symmetry, chords, perpendicular bisectors, circumcircles, arc angles and cyclic quadrilaterals.
Circle and Symmetry
A circle is the locus of points at a fixed distance from a centre. Every diameter is a line of reflection symmetry, and the circle has rotational symmetry through every angle.
Circles Through Points
Infinitely many circles pass through two points, with centres on the perpendicular bisector. Three non-collinear points determine one unique circle: the circumcircle.
Chords and Central Angles
Equal chords subtend equal angles at the centre, and equal central angles subtend equal chords.
Midpoint and Perpendicular Theorems
The line from the centre to the midpoint of a chord is perpendicular to the chord. Conversely, a perpendicular from the centre bisects the chord.
Chord Length and Distance
Equal chords are equidistant from the centre, and equidistant chords are equal. Of two unequal chords, the longer chord lies closer to the centre.
Arc Angles
Angles subtended by the same arc in the same segment are equal. A diameter subtends 90° at any point on the circle.
Concyclicity and Cyclic Quadrilaterals
If segment AB subtends equal angles at C and D on the same side, A, B, C and D are concyclic. In a cyclic quadrilateral, opposite angles are supplementary, and the converse is also true.
How to Choose the Right Theorem
| Clue in the problem | Useful result |
|---|---|
| Three non-collinear points | Unique circumcircle |
| Equal chords | Equal central angles / equal distances from centre |
| Perpendicular from centre to chord | Chord is bisected |
| Midpoint of chord joined to centre | Line is perpendicular to chord |
| Chord length and centre distance | Use right triangle and Pythagoras |
| Arc angle at centre and circle | Central angle is double |
| Diameter subtends an angle | Angle is 90° |
| Opposite angles add to 180° | Quadrilateral is cyclic |
• Treating every chord as a diameter, • Measuring chord distance along a slanted line instead of perpendicularly, • Forgetting that perpendicular from the centre bisects the chord, • Using the minor central angle when the problem refers to a major arc, • Forgetting that the point receiving an arc angle must lie outside that chosen arc, • Assuming four points are concyclic without a valid condition, • Mixing up a theorem with its converse,
Guided Practice
Problem
A chord is 5 cm from the centre of a circle of radius 13 cm. Find its length.
- 1.The perpendicular from the centre bisects the chord.
- 2.Let half the chord be x.
- 3.13²=5²+x².
- 4.169=25+x², so x²=144 and x=12.
- 5.Full chord=24 cm.
Problem
A circle has diameter 26 cm and chord length 24 cm. Find the chord's distance from the centre.
- 1.Radius=13 cm.
- 2.Half-chord=12 cm.
- 3.Let perpendicular distance be d.
- 4.13²=d²+12².
- 5.169=d²+144.
- 6.d²=25, so d=5 cm.
Problem
An arc subtends 70° at the centre. Find the angle on the remaining circle.
- 1.Central angle=2×inscribed angle.
- 2.Inscribed angle=70°/2=35°.
Problem
ABCD is cyclic, ∠A=75° and ∠B=110°. Find ∠C and ∠D.
- 1.∠A+∠C=180°, so ∠C=105°.
- 2.∠B+∠D=180°, so ∠D=70°.
Problem
A chord of length 16 cm lies 6 cm from the centre. Find the radius.
- 1.Half-chord=8 cm.
- 2.The perpendicular, half-chord and radius form a right triangle.
- 3.r²=6²+8²=36+64=100.
- 4.r=10 cm.
Practice Problems
- A chord is 5 cm from the centre of a circle of radius 13 cm. Find its length.
- An arc subtends 70° at the centre. Find the angle it subtends on the remaining circle.
- The diameter is 26 cm and a chord is 24 cm long. Find the chord's distance from the centre.
- A circle has radius 15 cm and a chord is 9 cm from the centre. Find the chord length.
- Prove that the perpendicular bisector of a chord passes through the centre.
- AB is a diameter and C lies on the circle. Find ∠ACB and justify it.
- ABCD is cyclic. If ∠A=75° and ∠B=110°, find ∠C and ∠D.
- In cyclic PQRS, ∠P=(2x+10)° and ∠R=(3x−20)°. Find x, ∠P and ∠R.
- A chord of length 16 cm is 6 cm from the centre. Find the radius.
- Show that a rectangle is the only parallelogram that can be cyclic.
- Show that if a rectangle is inscribed in a circle, its diagonals intersect at the centre.
- Explain why no chord can be longer than the diameter.
- A regular hexagon is inscribed in a circle of radius r. Find its side length and distance of each side from the centre.
- Two parallel chords of lengths 10 cm and 24 cm lie on the same side of the centre, 7 cm apart. Find the radius.
Quiz
What is the locus of points equidistant from two fixed points A and B?
Which chord is longest in a circle?
If a central angle subtended by an arc is 120°, what is the angle on the remaining circle?
If opposite angles of a quadrilateral are supplementary, what follows?
A perpendicular from the centre of a circle to a chord does what?
Check that you can define every part of a circle, construct a circumcircle, use chord-angle theorems, move between midpoint and perpendicular properties, calculate chord lengths using a right triangle, apply the central-angle theorem, and recognise or prove cyclic quadrilaterals.
Key Takeaways
• Circle geometry grows from equal radii and symmetry. • Perpendicular bisectors determine circle centres. • Chords, their central angles and their distances from the centre are tightly linked. • The perpendicular from the centre is the key tool in chord-length problems. • Arc angles connect central geometry with angles on the circle. • Cyclic quadrilaterals are characterised by supplementary opposite angles.
This completes the chapter. Continue by practising how to identify the correct theorem from the information given in a diagram.
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Concyclicity and Cyclic Quadrilaterals
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