Circle Properties: O Level Exam Preparation Guide

This revision note aims to help your child revise the Circle Properties and become proficient in solving the questions.

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What is a Circle?

A circle is a round-shaped 2-dimensional figure that has the same fixed distance from the center.

This distance is the radius of the circle.

The centre is the fixed point in the middle of the circle.

circle centre and radius

Terminology for Circles

circle arc, chord, segment

Arc        : A segment along the circumference of a circle.

Chord    : A line segment with both endpoints on a circle.

Segment: The area bounded by a chord and the included arc.

circle radius, diameter, tangent

Radius   : A line segment that joins the centre of a circle with any point on its circumference.

Diameter: A straight line segment passing through the centre of a circle and has its endpoints on the circle.

Circumference: is the distance around a circle.

Tangent: A line that touches a circle at only one point.

Secant: A line that touches the circle at two distinct points.

Semi-circle: a half of a circle with one curved edge (circumference) and one straight edge (diameter).

Properties of a Circle

In this section, we introduce every circle formula that you might need to use in your O Level exams.

Symmetry properties

1.  Equal chords are equidistant from the centre

circle radius, diameter, tangent

2.  The perpendicular bisector of a chord passes through the centre

circle symmetry properties, circle diagram for perpendicular bisector of a chord

3.  Tangents from an external point are equal in length

If TA and TB are tangents from T to a circle centre O
then TA = TB

circle symmetry properties, circle diagram with tangents from an external point

 

4. The line joining an external point to the centre of the circle bisects the angle between the tangents

If TA and TB are tangents from T to a circle centre O

circle symmetry properties, circle diagram with angle bisector between tangents

Angle properties

1.  Angle in a semicircle is a right angle

circle angle properties, circle diagram of semicircle and right angle

2.  Angle between tangent and radius of a circle is a right angle

If OA is the radius, BC is the tangent,

A is the point of contact,

then OA is perpendicular to BC

circle angle properties, circle diagram of angle between tangent and radius with right angle

3.  Angle at the centre is twice the angle at the circumference subtended by the same arc.

circle angle properties, 4 circle diagrams on angle at the centre and angle at the circumference

4.  Angles in the same segment are equal

circle angle properties, circle diagram of angles in the same segment

5.  Angles in opposite segments are supplementary

angle in cyclic quad

Types of Circle Properties Exam Questions (With Solutions)

Example 1

The figure shows a circle with centre O. AB = 8 cm and OM = ON = 3 cm.

(i)    Find the length of CN.

(ii)   Find the radius of the circle.

(iii)  Calculate  angle OCN.

circle diagram for exam question example 1

Solutions

circle diagram for exam question example 1

Example 2

Given the diagram, find the value of x and y.

circle diagram for exam question example 2

Solutions

solutions for circle properties example 2

Example 3

In the diagram, TA and TB are tangents to the circle. O is the centre of the circle.

Angle BTA = 44º.

Find the value of

a) ∠OAT
b) ∠AOB
c) ∠OBA
d) ∠ACB

circle diagram for exam question example 3

Solutions

circle diagram for exam question example 3

 

Example 4

In the diagram below, not drawn to scale, O is the centre of the circle.

TA and TB are tangents to the circle.  OA and OB are radii of the circle.

TOC is a straight line. OB intersects FH at G.  BH is parallel to TC.

Angle OTB = 36º

Find

(i) ∠AOF

(ii) ∠BHC

(iii) ∠OFG

circle diagram for exam question example 4

Solutions

circle diagram for exam question example 4

For more Secondary Math revision tips, check out our free math eBooks and test papers for secondary school students!

Last Minute Revision for O-Level Math?

Check out our exam guide on other topics!

Before you go, you might want to download this entire revision notes in PDF format to print it out, or to read it later. 

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To Your Child’s Success,
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Mr Tan Kok Heng
NIE Trained
Math Specialist
Jimmy Maths and Grade Solution Learning Centre

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