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O-Level The properties of the circle
What the O-Level syllabus expects for The properties of the circle, and how to practise it.
What the syllabus expects
- the symmetry facts of a circle: chords of equal length sit equally far from the centre; the perpendicular bisector of any chord passes through the centre; two tangents drawn from a single external point have the same length; and the join from that external point to the centre splits the angle between those tangents in two
- the angle facts of a circle: a semicircle subtends a right angle; a radius meets a tangent at 90°; the angle formed at the centre is twice whatever angle rests on the same arc at the circumference; two angles drawn within one segment match each other; and a pair of angles lying in opposite segments together make 180°
Why the angle at C is half the angle at O
The diagram is not to scale. The central angle AOB and the angle at the circumference ACB both use A and B as their endpoints. Because C lies on the major arc AB, angle ACB looks across the circle to the minor arc AB.
For angles standing on that same minor arc, the angle at the centre is twice the angle at the circumference. The labels matter: changing which arc the angle subtends can change the answer.
Worked examples
Example 1 (4 marks)
A and B lie on a circle with centre O. The smaller angle AOB is 124°, and C lies on the major arc AB. (a) Find angle ACB. (b) A tangent is drawn at A. Find the acute angle between the tangent and chord AB. Give a geometrical reason for each answer.
Show the worked answer
Step 1 - identify the arc used by angle ACB. C is on the major arc AB, so angle ACB looks across to the minor arc AB. The central angle standing on that same minor arc is AOB = 124°. Step 2 - apply the angle-at-the-centre theorem. The angle at the centre is twice the angle at the circumference standing on the same arc. Therefore AOB = 2 × ACB ACB = 124° ÷ 2 = 62°. Step 3 - apply the alternate segment theorem. The acute angle between the tangent at A and chord AB equals the angle made by chord AB at the circumference in the alternate segment. That angle is ACB, so the required tangent-chord angle is also 62°. Check: 62° is acute, as requested, and doubling it returns the given central angle of 124°.
Common mistakes to avoid
- Halving an angle without checking that the central angle and circumference angle stand on the same endpoints and the same arc.
- Using the reflex angle 236° just because C is on the major arc. The angle at C looks toward the opposite, minor arc, so this question uses the smaller central angle 124°.
- Writing only 62° and losing a reason mark. Name the angle-at-the-centre theorem for part (a) and the alternate segment theorem for part (b).
- Using radius perpendicular to tangent to answer part (b) directly. That fact gives 90° between OA and the tangent, not the requested angle between the tangent and AB.
Related circle-theorem skills
Angles, triangles and many-sided figures · The theorem of Pythagoras and trigonometry · Congruence alongside similarity
More O-Level Elementary Maths (E-Maths) topics
Whole numbers and the operations performed on them · Ratio and proportion · Percentage · Rate and speed · Algebraic expressions and formulae · Functions together with their graphs · all of O-Level Elementary Maths (E-Maths)