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.
![Circle Theorems](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fbab92f86-e1ea-4fc6-b946-225b025fbf09-file.png&w=1920&q=75)
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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.
![Angle in a Semi-Circle](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F71ae1129-b40f-486f-abbd-cfa504a68e98-file.png&w=828&q=75)
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
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,
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°.
![Angle Between Tangent and Radius](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Fa3ae4427-20e3-425a-b751-6f1135e55041-file.png&w=1200&q=75)
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 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, .
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.
![Angle at the Centre](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2Faf4f1a15-50ef-4fce-bd81-461c075d5a3c-file.png&w=1080&q=75)
Example:
Given a circle with centre O and points A, B, and C on its circumference, forming angles at the centre and 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. subtends arc AB, as does .
3. By the theorem,
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 , given that .
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, .
3. Recognizing given as , place it within the right-angled triangle ABC.
4. Apply the angle at the centre theorem: (angle at the centre) is twice (angle at the circumference), so
5. To find , note that it includes and two right angles at points A and B (since the tangent at a point creates a right angle with the radius).
6. Therefore,