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Circle Theorems
Mathematic ProjectSession 2011-12
Made ByRenu Bhati
Class 9th D
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ACircle features.
the distance aroundthe Circle
its PERIMETER
Diameter
the distance acrossthe circle, passingthrough the centre ofthe circle
Radius
the distance fromthe centre of the circleto any point on thecircumference
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ACircle features.
a line joining twopoints on thecircumference.
chord divides circleinto two segments
part of thecircumference of acircle
MajorSegment
MinorSegment
a line which touchesthe circumference atone point only
From Italian tangere,to touch
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Properties of circles When angles, triangles and quadrilaterals
are constructed in a circle, the angles
have certain properties
We are going to look at 4 such properties
before trying out some questions together
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AnANGLE
on a chord
An angle that sits on a
chord does not change asthe APEX moves around
the circumference
as long as it stays
in the same segment
We say Angles
subtended by a chordin the same segment
are equal
Alternatively Angles
subtended by an arcin the same segment
are equal
From now on, we will only consider the CHORD, not the ARC
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Typical examplesFind angles a and b
Imagine the Chord
Angle b = 28
Imagine the Chord
Angle a = 44
Very often, the exam
tries to confuse you by
drawing in the chords
YOU have to see the
Angles on the same
chord for yourself
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Angle at the centre
Consider the two angles
which stand on this
same chord
What do you notice
about the angle at the
circumference?
It is half the angle at the
centre
We say If two angles stand on the same chord,
then the angle at the centre is twice the angle at
the circumference
A
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Angle at the centre
We say If two angles stand on the same chord,
then the angle at the centre is twice the angle at
the circumference
Its still true when we move
The apex, A, around the
circumference
AAs long as it stays in the
same segment
136
272
Of course, the reflex angle
at the centre is twice the
angle at circumference too!!
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Angle at Centre A Special Case
When the angle stands
on the diameter, what is
the size of angle a?
aa
The diameter is a straight
line so the angle at the
centre is 180
Angle a = 90
We say The angle in a semi-circle is a Right Angle
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ACyclic Quadrilateral
is a Quadrilateral
whose vertices lie on the
circumference of a circle
Opposite angles in a
Cyclic Quadrilateral
Add up to 180
They are supplementary
We say
Opposite angles in a cyclic quadrilateral add up to 180
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Questions
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Could you define a rule for this situation?
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Tangents When a tangent to a circle is drawn, the
angles inside & outside the circle have
several properties.
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1. Tangent & RadiusA tangent is perpendicular
to the radius of a circle
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