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.
