Archive for the 'Circle Theorems' Category

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GS Checkpoint 4


Using the all the circle theorems that you’ve learnt, complete the worksheet attached.  Show all your working out in your workbook.



Circle Theorem 5

Before looking at Circle Theorem 5, some other properties of circles that we should know first:

  1. The perpendicular from the centre to the chord bisects the chord
  2. The angle between the tangent and radius is 90°
  3. Tangents from a point outside the circle are equal in length



Circle Theorem 5

The angle between the tangent and chord at the point of contract is equal to the angle in the alternate segment.


For example:  What is the size of angle x and angle y?



angle x = 60°

angle y = 80° (The angles in a triangle add up to 180°)


Now, complete the self-testing activity.  Remember to record your responses in your workbook.

Circle Theorem 4

cyclic quadrilateral is a quadrilateral whose vertices all touch the circumference of a circle. The opposite angles add up to 180o.

In the cyclic quadrilateral below, angles A + C = 180o, and angles B + D = 180o.  

Question 1

What is the size of:  

  1. angle A?
  2. angle B?

The Answer

  1. Did you get 120°
    Well done. A + 60° = 180°, so A = 120°.
  2. Did you get 40°
    Well done. B + 140° = 180°, so B = 40°.
Therefore ….

Opposite angles in a cyclic quadrilateral add up to 180°.


Practice Questions

Question 1


Question 2


Circle Theorem 3

We have just seen that the angle at the centre of a circle is double the size of the angle at the edge, and that angles in the same segment are equal.  

Look at the diagram below:  


The angle at the centre (AOB) is twice the angle at the circumference (APB). 
As AOB is 180°, it follows that APB is 90°.  

AOB is the diameter, so it follows that the angle in a semicircle is always a right angle:  



The angle subtended by a diameter is a right angle (90°).


Circle Theorem 2

Angles subtended at the circumference, in the same segment, by the same chord, are equal.


Example 1:


Example 2:



Practice Questions

1.  Find the pronumerals for Figure 1 and 2.

2.  Using the result from Circle Theorem 2, prove the additional result that x = z in Figure 3 below.

Circle Definitions

In Level 5, we are going to look at several important properties and theorems of circles.

First, we will find out some properties of circles:


Circle Definitions


a.  Divide your workbook into 10 columns, as shown below:

b.  Write down a key word at the top of a ‘box’.  In your own word, write down the meaning and draw a picture/symbol/diagram to represent it.  The first one has been done for you:



  • radius
  • circumference
  • diameter
  • chord
  • minor segment and major segment
  • arc, major arc and minor arc
  • subtend
  • angle subtended by the chord at the centre
  • angle subtended by the chord at the circumference
  • tangent line

Circle Theorems 1

Previous, we look at several important properties of circles.  


Now, we will look at the first theorem of circles.


Circle Theorem 1

Go to website.  This activity will proof circle theorem 1.

When log on to the website, click on point P or point O and drag.  Look at the value for angle Y and angle δ on the left hand side.  Find the relationship between angle Y and angle δ .




The angle subtended by the chord at the centre is twice the value of the angle subtended by the same chord at the circumference.

In other word

The angle subtended at the circumference is half the angle subtended at the centre by the same arc, i.e.

angle X = 2 x angle Y


Try these: