Circle theorems are a crucial aspect of geometry, offering insights into the relationships and properties within circles. These theorems enable us to solve a variety of problems and understand the geometric principles that govern circles. In this section, we will delve into key circle theorems, including the angle in a semicircle, the angle between a tangent and radius, and the relationship between angles at the centre and the circumference. Through examples and equations, we aim to provide a comprehensive understanding of these concepts.

Image courtesy of EzyMaths

Angle in a Semicircle

The theorem states that the angle in a semicircle is 90°. This means that any angle formed at the circumference of a circle by a diameter is a right angle.

Image courtesy of Third Space Learning

Example:

Consider a circle with centre O and diameter AB. If C is any point on the circumference such that ACB forms a triangle, then $∠ACB = 90°.$

Solution:

1. Draw circle with centre O and diameter AB.

2. Mark point C on the circumference, ensuring ACB is a straight line.

3. By the theorem, $∠ACB = 90°.$

Angle Between Tangent and Radius

This theorem highlights that the angle between a tangent to a circle and the radius drawn to the point of contact is 90°.

Example:

Let a circle with centre O and a tangent T at point P on the circle's circumference, with OP being the radius. The angle $∠OPT$ formed is always 90°.

Solution:

1. Draw circle with centre O and tangent T at point P.

2. Draw radius OP to the point of tangency P.

3. By the theorem, $∠OPT = 90°$.

Angle at the Centre

The angle at the centre of a circle is twice the angle at the circumference when both angles subtend the same arc.

Example:

Given a circle with centre O and points A, B, and C on its circumference, forming angles $∠AOB$ at the centre and $∠ACB$ at the circumference, where both angles subtend the same arc AB.

Solution:

1. Draw circle with centre O and points A, B, and C on the circumference.

2. $∠AOB$ subtends arc AB, as does $∠ACB$.

3. By the theorem, $∠AOB = 2∠ACB.$

Applying Circle Theorems

Understanding and applying these theorems can solve complex geometric problems involving circles. Let's consider a problem that combines these theorems.

Problem:

Given a circle with centre O, diameter AB, and a point C on the circumference, a line from C forms a tangent at point T. Find the angle $∠ATB$, given that $∠ACB = 30°$.

Solution:

1. Identify and mark all known angles and lines, including the tangent T at point C and diameter AB.

2. Using the angle in a semicircle theorem, $∠ACB = 90°$.

3. Recognizing $∠ACB$ given as $30°$, place it within the right-angled triangle ABC.

4. Apply the angle at the centre theorem: $∠AOB$ (angle at the centre) is twice $∠ACB$ (angle at the circumference), so $∠AOB = 2 × 30° = 60°.$

5. To find $∠ATB$, note that it includes $∠AOB (60°)$ and two right angles at points A and B (since the tangent at a point creates a right angle with the radius).

6. Therefore, $∠ATB = 60° + 90° + 90° = 240°.$