Circle Properties

Circle Properties

Introduction
In Senior Two, you identified various parts of a circle and learnt how to state and use the formulae for area and circumference of a circle. Building on that prior knowledge, you will discover and learn more about circles and their properties, and how you can use them to solve problems. Circles are fundamental in geometry because of their richness in applicability in daily life. For example, the properties of circles are widely applied in construction of buildings and houses, and in designing playgrounds. These properties are also used in designing tyres for cars and bicycles. Have you ever wondered why tyres are circular but not any other shape?
13.1 Identifying the Arc, Chord, Sector and Segment
From the activities given below, you will explore the circle and identify other parts of the circle.

Activity 13.1 (b)
(Work in groups)
(a) Draw a circle.
(b) Draw a line from the centre of your circle to any point on the circumference.
(c) Draw another line from the centre to another point on the same circumference.
(d) Shade the area enclosed by the two lines.
(e)Explain your observation to the class.
(f)Identify any two points on the circumference, below or above the centre. Draw a line joining these two points.
(g)Is the line passing through the centre to any point on the circumference different from that in (f)? If yes, what is the difference?

(a) Describe a segment.
(b) How does the area of a segment relate with the area of a sector?
(c) Compare your notes with other groups.
13.2 Relating Angles made by an Arc at the
Circumference and Centre
Under this subtopic, you will investigate the relationship between angles made by an
arc at the circumference and centre.
Relating angles made by an arc at the circumference
Activity 13.2 (a) (Work in groups)
(a) Draw a circle on a piece of paper.
(b)Mark two points, A and B, on the circumference of your circle.
(d)Use points A and B to form any three angles at the circumference, in the same segment, by drawing lines.
(e)Measure and record the angles formed in (c).
State your observation?

Relating angles made by an arc at the circumference and centre
Activity 13.2(b) (Work in groups)
(a) Draw a circle and mark the centre as O.
(d)Draw any two radii and mark their endpoints as A and B.
(e)Form the chord on the circle by joining the two points.
(f)Mark any point, C on the circumference.
(e)Connect A to C, and B to C.
Measure the angles AOB and ACB.
Write your observation.

13.3 Determining the Tangent, Chord and Angle Properties of a Circle
You shall discover how to tangent and chord relate to the angles and the length through their properties.
The tangent and angle properties of the circle
Activities 13.3(a) and 13.3(b) will be instrumental in helping us explore about the tangent and the angle properties of the circle.
Activity 13.3(a) (Work in groups)
(a) Draw a circle with centre O.
(b) Draw any straight line that just touches the circle at any one point. How would you call such a line.
(c) Choose any point on the circumference and label it x.
(d) Draw a line connecting point x to the centre O.
(e) Draw a line outside the circle on the circumference passing through point x.
(f) Measure the angle between the two lines meeting at point x.
(g) What conclusion can you make about the radius, the line passing through the point and the angle?

The chord and angle properties of a circle
You are required to explore the chord and the angle properties of the circle through
the following activities.
Activity 13.3(d) (Work in groups)
(a) Draw a straight line having its ends on the circumference of the circle. How would you call such a line?.
(d)Inscribe similar figures in your circle as shown in the figure A
below.
Measure lengths: AB, BO, BC, AC, AO, and OC.
Measure angles y and x. What conclusion can you make about the angles?
c
x

Activity 13.3(e) (Work in groups)
(b) Draw a circle and label the centre as O.
(d)Draw the diameter and label the end points of the diameter as C and D.
(c)Choose any point Y on the circumference.
(e)Connect point C to point Y and point D to point Y.
Measure angle C YD.
(f)Mark any point X on the circumference and measure angle CXD.
(e)Briefly explain the relationship between the angles subtended by CD on the circumference.

13.5 Finding the Length of the Common Chord for two Intersecting Circles
In Senior Two, you drew Venn diagrams. Do you recall how to draw them? How
many points of intersection does a Venn diagram of 2 sets have?
Intersection of circles
Activity 13.5(a) (Work in groups)
Suggested Materials: a pair of compass, pencil, ruler
(a) Draw two intersecting circles of your choice.
(b) Show the common chord for the two intersecting circles,
(c) Present your findings to the rest of the class.
Activity 13.5(b) (Work in groups)
Suggested materials: a pair of compass, pencil, ruler.
For each case identified in Activity B.5(a);
(a) draw any two intersecting circles of different radii.
(b) draw a line joining the centres of the two circles drawn in
(c) measure the length of the line drawn in (b) above,
(d) what would you say about the two radii of the circles and the length
measured in (c)?
(e) share your findings with the rest of the class.
Activity 13.5(c) (Work in groups)
Suggested materials: a pair of compass, pencile ruler.
(a) Draw any two circles which intersect at two points.
(b) Shade the common area.
(c) Draw a segment of each circle in (o) to ensure that the area in (b) is enclosed.
(d) How dces the common area relate with the areas of the two segments in (c)?
Finding the length of the common chord for two intersecting circles
We would like to investigate whether there is a relationship between the radii of any
two intersecting circles and their common chord This will be through activity 73.5(d).
Activity 13.5(d) (Work in groups)
(a) Draw any two circles which intersect to form a common chord.
(b)Draw lines joining the centres of the circles in (a) to both ends of the chord.
(c)Measure the angles at the centres in (b) and the length of the common chord.
(d)What do you conclude about the length of the common chord, the radius of any of the two circles and the angle at the centre of that circle.
(e)Share your responses with the rest of the class.

13.6 Calculating Areas of Sectors and Segments
Sectors and segments are part of the area of a circle.
Calculating the area of sectors
A sector is the area of a circle bounded by 2 radii and the corresponding arc.
Activity 13.6(a)
(Work in groups)
Suggested materials: pencil, circular model, papers
(b) State the formula for the area of a circle with radius r as learnt in Senior Two.
(c)Use any circular object from your environment to draw a circle.
(d)Measure 600 on the circle and shade it. How would you call the shaded part?
(e)Determine the area of the shaded part (sector).
Present your responses to other groups.

Topic Summary
In this topic, you have learnt that:
1) A circle has different parts, and you have learnt how to identify them.
2)A perpendicular dropped from the centre of a circle divides the chord into two equal parts.
3)The radius of a circle is always perpendicular to the tangent at the point of
contact with the circle.
4)Angles formed by the same arc or chord on the circumference of a circle in the same segment are equal.
5)The angle at the circumference in a semi-circle is always 900
6)The angle subtended at the centre of a circle is twice that subtended at the circumference.