How do I recall the central-angle formulas?

Cut the isosceles triangle along its height: 2r sine of the half angle is the chord, r cosine is the distance.

Why is the tangent-chord angle not halved?

It equals the inscribed angle in the alternate segment directly; only a central angle is double.

S5 Maths: Properties of Circles (10 Challenge Items)

2026-11-12

Ten combined items: longer chords lie nearer the centre, central angles give chords and distances through sine and cosine, the tangent-chord angle equals the alternate segment angle, and a chord tangent to a concentric circle.

Multi-step reasoning and formula switching　The challenge paper chains the theorems: compare centre distances from chord lengths, use 2r sin of half the angle for a chord and r cos of half the angle for the distance, then handle the tangent-chord angle, which equals the angle in the alternate segment, and chords tangent to a concentric circle. Name the theorem used in each answer.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Challenge 9 · Error redo 1
Chapter	Chapter: S5 Maths: Properties of Circles – Chords, Arcs and Tangents
Level	S5 Maths

02　Questions (10 items)

ChallengeIn a circle of radius 13 cm, chord AB is 24 cm and chord CD is 10 cm. Which is nearer the centre, and by how much?
ChallengeTwo parallel chords of 24 cm and 10 cm in a circle of radius 13 cm lie on opposite sides of the centre. Find the distance between them.
ChallengeA circle of radius 6 cm has a central angle AOB of 120 degrees. Find chord AB to 2 decimal places and the distance from the centre to AB.
ChallengeA circle of radius 10 cm has a central angle of 90 degrees. Find the chord and the distance from the centre to it, to 2 decimal places.
ChallengeTangents PA and PB from P make angle APB of 64 degrees. Find angle AOB at the centre.
ChallengeA tangent makes 65 degrees with chord AB. Find the angle in the alternate segment.
ChallengeTwo concentric circles have radii 13 cm and 5 cm. A chord of the large circle is tangent to the small one. Find the chord length.
ChallengeTangents PA and PB are each 15 cm and OP is 17 cm. Find the radius.
ChallengeA circle of radius 8 cm has a central angle of 60 degrees. Find the chord and describe the triangle formed.
Error redoRetry: a student says the tangent-chord angle equals the central angle. If the angle is 50 degrees, what are the inscribed and central angles?

03　Answers and one-line notes

#	Answer	One-line note
1	AB is nearer: 5 cm from the centre against 12 cm for CD.	The longer chord is nearer the centre: 5 cm against 12 cm.
2	17 cm.	Distances are 5 and 12; opposite sides means adding them.
3	AB is 6 root 3, about 10.39 cm, and the centre distance is 3 cm.	The chord is twice 6 sine 60, about 10.39, and the centre distance is 6 cos 60, which is 3.
4	The chord is 10 root 2, about 14.14 cm, and the distance is 5 root 2, about 7.07 cm.	The chord is 20 sine 45 and the distance is 10 cosine 45.
5	Angle AOB is 116 degrees.	Two right angles at the points of contact leave 116 degrees at the centre.
6	65 degrees.	The tangent-chord angle equals the angle in the alternate segment, so it is not halved.
7	24 cm.	Tangency fixes the distance at 5 cm, so half the chord is 12.
8	8 cm.	The radius is the root of 289 minus 225, and OP is the hypotenuse.
9	The chord is 8 cm and triangle OAB is equilateral.	With two radii of 8 and a 60 degree angle the third side is also 8, so the triangle is equilateral.
10	The inscribed angle is 50 degrees and the central angle is 100 degrees.	The tangent-chord angle equals the inscribed angle, and the central angle is twice that.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.