Download Circle theorems

GCSE Mathematics (1-9) 2015
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G10Theory
Geometry - Circle theorems - 10
apply and prove the standard circle theorems concerning angles, radii,
tangents and chords, and use them to prove related results
1
Angle at the centre
The angle at the centre is twice the angle at the
circumference
1.1
Proof
See second diagram
AB = BC (both
radius of circle)
So triangle ACB is isosceles
So angle CAB = angle BCA = w
angle ABC = 180 -2y (angle sum in triangle is 180)
Similarly
angle ABD = 180 - 2z
so CBD = ABC + ABD = 360- 2(y+z)
so w = 360-CBD = 2(y+z) (angle sum at a point = 360 )
GCSE Mathematics (1-9) 2015
2
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Angle on a diameter
The angle on a diameter is 90 degrees
2.1
Proof
This follows from 'angle at the centre'
If the angle at the centre = 180 (straight line) then
the angle at the circumference is 90
3
Angles in the same segment
Angle ABC = angle ADC
3.1
Proof
AOC = 2 ABC ( angle at centre )
AOC = 2 ADC ( angle at centre )
So ABC = ADC
4
Cyclic quadrilateral
A cyclic quadrilateral has its vertices on a circle, as shown
Opposite angles add up to 180
a+b=180
4.1
Proof
See second diagram
AO = BO ( both radius of circle)
So triangle AOB is isosceles.
So angle OAB = angle ABO
and similarly for the other vertices.
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GCSE Mathematics (1-9) 2015
How To Do it
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Angles in a quadrilateral add up to 360
so
A
x w
2x+2y+2z+2w = 360
B
so
(x+w) + (y+z) = 180
x
y
O
so
DAB = DCB
y
z
also
(x+y) + (w+z) = 180
so
ABC + ADC = 180
5
Tangent is perpendicular to radius
If BC is a tangent, angle BAO is a right angle.
5.1
Proof
Suppose its not, and that instead OE is perpendicular to
the tangent BC.
The perpendicular is the shortest distance to a point.
So OE < OA
So OE < radius
But OE = OD + DE = radius + DE
So OE < radius is impossible
So the initial assumption is false
C
w
z
D
GCSE Mathematics (1-9) 2015
6
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Two tangents equal
AB = CB
6.1
Proof
Triangles OAB and OCB are congruent ( RHS - the
side is the radius and the hypotenuse is common )
So AB = CB
7
Alternate segment
a=b
The line is a tangent to the circle.
It is the 'alternate' segment because a is in the
other segment from b : the alternate segment.
7.1
Proof
OB = OC (both radius)
so OBC is isosceles
so CBO = OCB
and similarly for the other triangles
OAD is a right angle (tangent is perpendicular)
So
CAD = 90 - z
2x+2y+2z=180 (angles in a triangle)
x+y+z=90
x+y=90-z
so CAD = CBA
a
b
GCSE Mathematics (1-9) 2015
8
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Perpendicular bisects chord
AB = BC
8.1
Proof
Triangles OBC and OBA are congruent ( RHS : hypotenuse is
radius, side is common )
So BC = AB
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