Circle Theorems

1
Circle Theorems

• Identify a tangent to a circle
• Find angles in circles
• Tangents
• Angles in a semicircle
• Cyclic quadrilateral
• Major and minor segments
• Angles at the circumference/centre
• Alternate segments

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(1) Tangents

• A tangent to a circle is a straight line that
  touches the circle at one point.

Not tangents
Drawing a radius to any tangent to a circle
always produces a 900 angle
Tangents

• The point where the tangent touches the circle is
  called the point of contact.

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The angles in circles
One side is the radius of the circle One side is
a tangent to the circle
For these triangles
Right angle
Right angle
Since these two lines meet at right angles- these
are right angle triangles!
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Finding angles
Remember Angles in a triangle add up to 1800
Once you can spot the right angles triangles- you
can find missing angles
900
32º
a
d
d
900
320 900 a 1800
d d 900 1800
32 90 58 180
45 45 90 180
a 580
d 450
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Angles in a semi circle
Use the diameter of a circle as the base of a
triangle
Any triangle you make using this base will form
a right angled triangle
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(2) Two tangents
Two tangents will create Isosceles Triangles
You need to identify Equal pairs of lines
(length) Equal pairs of angles
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Two tangents.
Use the right angle (tangent to radius) to
calculate the 200
Use Isosceles triangle to show 700 angles
700
200
40º
1400
200
700
Since angles in a triangle add up to 1800 show
the 1400
Use the isosceles triangle to show 200 angle
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(3) Cyclic Quadrilateral
In any cyclic Quadrilateral opposite corners sum
to 1800
So A C 1800 D B 1800
B
A
C
D
E
H
Any 4 points on a circle joined to form a
quadrilateral
So E G 1800 F H 1800
F
G
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Cyclic Quadrilateral Questions
What are the missing angles?
1400
900
900
1100
50º
40º
1300
70º
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(4) Major and minor segments
Any circle can be divided into two unequal parts.
These are called segments
You can use your chord as the base of a triangle
The larger part is the major segment. The smaller
part is the minor segment.
Minor segment
d
Major segment
Or
The line that divides the circle is called a
chord.
c
a
b
a b c d
Providing all the angles are in the same segments
they will be equal.
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(5) Angles at the centre/circumference
From a chord you can create a triangle at the
centre or at the circumference of the circle.
The angle at the centre is twice the size of the
angle at the circumference
At the centre
At the circumference
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Questions
b 600
2500
a 650
1100
C 1250
c
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Putting your rules together
Angles at a tangent 900
An isosceles triangle has two equal angles
40º
500
Angles in a triangle add up to 1800
500
2600
1000
Angle at the centre is twice that at the
circumference
400
Angles round a point add up to 3600
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(6) Alternate segments
Angles in alternate segments are equal
A
B
A
B