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Circle Theorems Identify a tangent to a circle Find angles in circles
TangentsAngles in a semicircleCyclic quadrilateralMajor and minor segmentsAngles at the circumference/centre Alternate segments
(1) Tangents A tangent to a circle is a straight line
that touches the circle at one point.
The point where the tangent touches the circle is called the point of contact.
Not tangents
TangentsDrawing a radius to any tangent to a circle always produces a 900 angle
The angles in circles
For these triangles:
One side is the radius of the circle
One side is a tangent to the circle
Since these two lines meet at right angles- these are right angle triangles!
Right angle Right
angle
Finding angles
32º
d
d
Once you can spot the right angles triangles- you can find missing angles
Remember: Angles in a triangle add up to 1800
900
900
a = 580
320 + 900 + a = 1800
a
32 + 90 + 58 = 180 d + d + 900 = 1800
45 + 45 + 90 = 180
d = 450
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
(2) Two tangents
Two tangents will create Isosceles Triangles
You need to identify:
Equal pairs of lines (length)
Equal pairs of angles
Two tangents.
40º
Use “Isosceles triangle” to show 700 angles
700
700
200
Use the right angle (tangent to radius) to calculate the 200
Use the isosceles triangle to show 200 angle
200
Since angles in a triangle add up to 1800 show the 1400
1400
(3) Cyclic Quadrilateral
A
B
C
D
Any 4 points on a circle joined to form a quadrilateral
In any cyclic Quadrilateral opposite corners sum to 1800
So:
A + C = 1800
D + B = 1800
E
FG
HSo:
E + G = 1800
F + H = 1800
Cyclic Quadrilateral Questions
70º
50º 40º
1300
What are the missing angles?
1100 900
1400
900
(4) Major and minor segments
Providing all the angles are in the same segments they will be equal.
Major segment
Minor segment
Any circle can be divided into two unequal parts. These are called segments
The larger part is the major segment.
The smaller part is the minor segment.
The line that divides the circle is called a chord.
You can use your chord as the base of a triangle
Or…
ab
c
d
a = b = c = d
(5) Angles at the centre/circumferenceFrom a chord you can create a triangle at the centre or at the circumference of the circle.
At the centre
At the circumference
The angle at the centre is twice the size of the angle at the circumference
Questions
130º
a
b
300
a = 650
b = 600
1100
2500
C = 1250c
Putting your rules together
40º
Angles at a tangent = 900
500
An isosceles triangle has two equal angles
400
Angles in a triangle add up to 1800
1000
Angle at the centre is twice that at the circumference
500
Angles round a point add up to 3600
2600
(6) Alternate segments
A
A
B
B
Angles in alternate segments are equal