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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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