# 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.

Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
This will be delivered to your email inbox. ## 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. ## Terminology for Circles

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.

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

#### 1.  Angle in a semicircle is a 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 #### 3.  Angle at the centre is twice the angle at the circumference subtended by the same arc. #### 4.  Angles in the same segment are equal #### 5.  Angles in opposite segments are supplementary ## 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. #### Solutions ### Example 2

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

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

This will be delivered to your email inbox.  Mr Tan Kok Heng
NIE Trained
Math Specialist
Jimmy Maths and Grade Solution Learning Centre

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