Why halve the chord first?

The theorem only says the foot bisects the chord; in the right triangle the hypotenuse is the radius and the legs are the centre distance and the half chord.

How should a tangent question start?

Join the radius to the point of contact, mark the right angle, then treat OP as the hypotenuse.

S5 Maths: Properties of Circles (10 Basic Items)

2026-11-12

Tutor Prep Guide · Primary Maths

Ten items on four essentials: the perpendicular from the centre bisects a chord, equal chords are equidistant, a tangent is perpendicular to the radius, and tangents from one external point are equal.

First use of the four theorems　Every item here rests on four theorems: the perpendicular from the centre bisects a chord; equal chords are equidistant from the centre; a tangent is perpendicular to the radius at the point of contact; and two tangents from one external point are equal. Draw the figure, mark the right angle and the half chord, then decide whether to use Pythagoras or the theorem directly.

01　Paper details (check answers afterwards)

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

02　Questions (10 items)

FoundationA circle of radius 5 cm has centre O. The perpendicular distance from O to chord AB is 3 cm. Find AM and AB, where M is the foot of the perpendicular.
FoundationA circle has radius 13 cm and chord AB = 24 cm. Find the distance from the centre to AB.
FoundationIn a circle of radius 10 cm a chord is 12 cm long. How far is it from the centre?
FoundationIn a circle of radius 9 cm the distance from the centre to a chord is 0. What is that chord and how long is it?
FoundationWhat is the angle between a tangent and the radius drawn to the point of contact?
FoundationThe radius is 7 cm and an external point P is 25 cm from the centre O. PT is a tangent at T. Find PT.
FoundationTwo tangents PA and PB are drawn from an external point P. If PA = 9 cm, find PB.
FoundationIn one circle, chord AB equals chord CD and both are 16 cm. How do their distances from the centre compare?
FoundationA circle of radius 17 cm has a chord 8 cm from the centre. Find the chord length.
Error redoRetry: a student claims the perpendicular from the centre to a chord need not bisect it. Is that right?

03　Answers and one-line notes

#	Answer	One-line note
1	AM = 4 cm and AB = 8 cm.	A perpendicular from the centre bisects the chord, so AB is twice AM; Pythagoras gives AM = 4.
2	5 cm.	Halve the chord to 12 first, then take the square root of 169 minus 144.
3	8 cm.	Half the chord is 6, and the root of 100 minus 36 is 8.
4	It is a diameter, 18 cm long.	A distance of zero means the chord passes through the centre, so it is a diameter of length 2r.
5	90 degrees, that is, they are perpendicular.	The tangent is perpendicular to the radius at the point of contact.
6	PT = 24 cm.	Angle OTP is a right angle, so PT is the root of 625 minus 49.
7	PB = 9 cm.	Two tangents from the same external point are equal because the two right triangles are congruent.
8	The two distances are equal.	Equal chords are equidistant from the centre.
9	30 cm.	Half the chord is 15, so the whole chord is 30; remember to double it.
10	No. The foot always bisects the chord, so AM equals MB.	The perpendicular from the centre always bisects the chord; only a line missing the centre may not.

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.

Book an S5 Maths tutor Tutor Registration →

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.