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Naming the parts of a circle

A circle is a set of points equidistant from its centre.A circle is a set of points equidistant from its centre.

The distance around the outside of a circle is called the circumference.

The radius is the distance from the centre of the circle to the circumference.

radius

circumference

The diameter is the distance across the width of the circle through the centre.

diameter

centre

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a sector is formed.

Arcs and sectors

An arc is a part of the circumference.

When an arc is bounded by two radii

arc

sector

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Right angles in a semicircle

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Calculating the size of unknown angles

Calculate the size of the labeled angles in the following diagram:

a = 37° (angles at the base of an isosceles triangle)

O37°

ab

c

de

b = 90° – 37° = 53° (angle in a semi-circle)

c = 53° (angles at the base of an isosceles triangle)

d = 180° – 2 × 53° = 74° (angles in a triangle)

e = 180° – 74° = 106° (angles on a line)

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The angle at the centre

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Calculating the size of unknown angles

Calculate the size of the labeled angles in the following diagram:

a = 29° (angles at the base of an isosceles triangle)

O

29°

41°

c

db

b = 180° – 2 × 29° = 122° (angles in a triangle)

c = 122° ÷ 2 = 61° (angle at the centre is twice angle on the circumference)

d = 180° – (29° + 29° + 41° + 61°) = 20° (angles in a triangle)

a

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Angles in the same segment

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Angles in the same segment

We have just seen a demonstration that the angles in the same segment are equal.

We can prove this result as follows:

A

B

D

C Mark the centre of the circle O and angle AOB.

angle ADB = ½ of angle AOB

and angle ACB = ½ of angle AOB

(the angle at the centre of a circle is twice the angle at the circumference)

angle ADB = angle ACB

O

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Calculating the size of unknown angles

Calculate the size of the labeled angles in the following diagram:

a = 90° – 51° = 39° (angle in a semi-circle)

O 44°

51°

c

b

b = 180° – (90° + 44°) = 46° (angles in a triangle)

c = 46° (angles in the same segment)

a

d

d = 51° (angles in the same segment)

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Angles in a cyclic quadrilateral

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Angles in a cyclic quadrilateral

We have just seen a demonstration that the opposite angles in a cyclic quadrilateral add up to 180°.

We can prove this result as follows:

B

D

AC

Mark the centre of the circle O and label angles ABC and ADC x and y.

The angles at the centre are 2x and 2y.

(the angle at the centre of a circle is twice the angle at the circumference)

2x + 2y = 360°y

x

O 2y

2x

2(x + y) = 360°

x + y = 180°

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Calculating the size of unknown angles

Calculate the size of the labeled angles in the following diagram:

a = 64° (angle at the centre)

128°db

b = c = (180° – 128°) ÷ 2 = 26° (angles at the base of an isosceles triangle)

d = 33° (angles at the base of an isosceles triangle)

e = 180° – 2 × 33° = 114° (angles in a triangle)

e

O

c

a

f

33° f = 180° – (e + c) = 180° – 140° = 40° (opposite angles in a cyclic quadrilateral)

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The tangent and the radius

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Two tangents from a point

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The perpendicular from the centre to a chord

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The alternate segment theorem

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