How do I remember same side and opposite side?

Draw the line through the centre perpendicular to both chords: same side means subtract, opposite sides means add.

Why use the half angle?

OP bisects angle APB into two congruent right triangles, so find the half angle and double it.

S5 Maths: Properties of Circles (10 Advanced Items)

2026-11-12

Ten harder items: parallel chords on the same side subtract and on opposite sides add, tangent length and OP are interchanged, and the tangent angle uses the sine of the half angle.

Parallel chords and tangent triangles　The advanced paper combines two situations: parallel chords relative to the centre, where same-side distances subtract and opposite-side distances add; and the tangent triangle, where any two of OP, the radius and the tangent length give the third, with OP bisecting the angle between the tangents.

01　Paper details (check answers afterwards)

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

02　Questions (10 items)

IntermediateA chord of 48 cm lies in a circle of radius 25 cm. Find the distance from the centre to the chord.
IntermediateIn a circle of radius 15 cm a chord is 9 cm from the centre. Find its length.
IntermediateIn a circle of radius 13 cm two parallel chords of 24 cm and 10 cm lie on the same side of the centre. Find the distance between them.
IntermediateThe same two parallel chords, 24 cm and 10 cm, now lie on opposite sides of the centre. Find the distance between them.
IntermediateA circle of radius 5 cm has an external point P 13 cm from the centre. Find the tangent length from P.
IntermediateWith radius 5 cm and OP = 10 cm, PA and PB are tangents. Find angle APB.
IntermediateA tangent from P is 20 cm long and the radius is 15 cm. Find OP.
IntermediateIn a circle of radius 13 cm, chord CD is 5 cm from the centre and is 24 cm long. Chord AB is also 5 cm from the centre. Find AB.
IntermediateA chord of 40 cm lies in a circle of radius 29 cm. Find its distance from the centre.
Error redoRetry: a student says the tangent length equals the radius. If OP = 25 cm and the radius is 7 cm, what is the tangent length?

03　Answers and one-line notes

#	Answer	One-line note
1	7 cm.	Half the chord is 24 and the root of 625 minus 576 is 7.
2	24 cm.	Half the chord is the root of 225 minus 81, which is 12, so the chord is 24.
3	7 cm.	The chords are 5 cm and 12 cm from the centre; on the same side, subtract to get 7.
4	17 cm.	On opposite sides you add: 5 plus 12 gives 17.
5	12 cm.	The tangent, the radius and OP form a right triangle, so the tangent is the root of 144.
6	60 degrees.	OP bisects the angle and the half-angle has sine 0.5, so each half is 30 degrees.
7	25 cm.	OP is the hypotenuse of a 15-20-25 triangle.
8	AB = 24 cm.	Chords equidistant from the centre are equal, so AB matches CD without further work.
9	21 cm.	Half the chord is 20 and the root of 841 minus 400 is 21.
10	24 cm.	The tangent is a leg, not the radius; the root of 625 minus 49 is 24.

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.