## Circle Geometry (Theorem 1 to 3)

## Parts of a circle

A **chord** of a circle is a straight line joining any two points on its circumference. A *diameter* is a *chord* that passes through the centre. A chord which doesn’t pass through the centre divide the circumference into 2 unequal arcs and also 2 unequal segments as shown below.

## Theorem 1

Angles subtended by the same chord are equal.

On the diagram, there are 3 angles, all of them from the circumference and subtended by the chord shown by the dashed line. Those 3 angles A, B and C are equal.

## Theorem 2

The angle from the center is twice the angle from the circumference, if they are subtended by the same chord.

On the digram, angle A is twice angle B and also twice angle C because angle A is from the centre.

## Theorem 3

The angle subtended by the diameter is a right angle.

In the digram, angle C is subtended by diameter AB and that is why it is a right angle.

## Examples

Find the lettered angles in the following

#### Example 1

Using Theorem 2, angle A = 73.19 × 2

= 146.38

#### Example 2

Angle B is a reflex angle at the centre and it is twice the obtuse angle at the circumference.

B = 123.09 × 2

= 246.18

#### Example 3

D = 100° × 2 *(theorem 2)*

= 200°

C = 360° – D *(angles at the centre add up to 360°)*

= 360° – 200°

= 160°

E = C ÷ 2 *(theorem 2)*

= 160° /2

= 80°

#### Example 4

We first deal with the right angled triangle on top because we know its 2 angles already. Lets find F, the third side.

F = 180° – 90° – 21° *(interior angles of a triangle add up to 180)*

= 69°

H = 2F *(Theorem 2)*

= 2 × 69°

= 138°

G = 360° – H *(Angles at a point add up to 360°)*

= 360° – 138°

= 222°

J = G/2 *(Theorem 2)*

J = 222°/2

J = 111°

#### Example 5

L = 40° *(Theorem 1)*

M = 40° *(Theorem 1)*

*angle L, angle M, and the 40° angle are all subtended by the same chord, even though the chord isn’t visible.*

*the diagram below shows the chord in dashed line.*

N = 32° *(Theorem 1)*

*angle N and the 32° angle are subtended by the same chord shown below in dashed line.*