Circle theorems

Circle theorems

Double AngleTriangles inside Circles

Angles connected by a chordTangents to a circle

Cyclic Quadrilaterals

2x

x

This is the ARC

o

Centre of Circle

The Angle x subtended at the centre of a circle by an arc is twice the size of the angle on the circumference subtended by the same arc.

2x

x

o

This is the ARC

Centre of Circle

Angle subtended at the Centre is twice the angle at the circumference

x

x

x

We are ALL EQUAL

This is the Arc

Minor Segment

Major Segment

o

A

B

CDx

180-x

If this angle was 600 then angle BCD would be 1800-600=1200

1200

Points which lie on the circumference of the same circle are called cyclic (or concyclic)points. A cyclic quadrilateral is a quadrilateralwith all its four corners (vertices) on thecircumference of the same circle.

T

A

B

O

TA=TB

Tangent

Tangent

Major Segment

Minor SegmentA B C

E

D

The Shaded Segment BED is called the alternate segment to the angle CBDThe angle between a tangent to a circle and a chord drawn through the pointof contact is equal to any angle subtended by the chord at the circumference in the alternate segment

Centre of Circle

Diameter

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The angle at the centre

25°

x

160°100°

60°

135°

90°

xx

xxx

12 3

64 5

Answers1) 502)1203)1804)505)67.56)80

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Double angle theorem

Right angles in a semicircle

60°

x

1 2 3

31 2

72°

x

x x

x y

y

x

100°

x30°

22°

y

Answers1) X=302)x=183)x=45

4)X=40 y=405)x=30 y= 1206)x=22 y=136

x

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Triangles inside circles

Angles in the same segment

25°x

12 3

645

y15°

yz

z

x

y

x

z

x

y

y

zx

25° 53°30°

z

y

x

80°17°

95°35°

40°

125°

15°

40°

10°

100°

Answers1) x=25 y=152)x=125 y= 40 z=153)x=10 y=70 z=1004)X=105 y=40 z=355)x=53 y= 30 z=726)x=85 y=80 z=17

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Angles connected by a chord(off the same arc)

The tangent and the radius

Two tangents from a point

40°xy

z

3

120°

x

4

140°

x

2

x35 °

1

y

z

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Tangents to a circle Answers:

1. x=552. x=403. x=50 y=50

z=404. x=60 y=60

z=30

Angles in a cyclic quadrilateral

x

yx

yx

y

95° 110° 54°

75°

20°

80°

x 2a

4b

15° 70° a

b

1

25°

yz

w

2 3

4 5Answers1) x=70 y=852)x=126 y=1053)x=100 y=1604)w=15 x=70 y=65 z= 255)a=60 b=36

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Cyclic Quadrilaterals

The alternate segment theorem