Circle Theorems Applied Problems
Circle theorems describe relationships between angles, chords, tangents, and arcs in a circle, forming the basis for solving complex geometric problems.
What You Need to Know
Key Concept Diagram
The angle at the centre is twice the angle at the circumference subtended by the same arc
Angles in the same segment are equal
The angle in a semicircle is always 90 degrees
Opposite angles in a cyclic quadrilateral sum to 180 degrees
The tangent to a circle is perpendicular to the radius at the point of contact
Key Vocabulary
Cyclic quadrilateral
A quadrilateral with all four vertices on a circle
Tangent
A line that touches a circle at exactly one point
Chord
A straight line segment connecting two points on a circle
Subtend
To be opposite to and delimit an angle or arc
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
An angle at the centre of a circle is 80 degrees. What is the angle at the circumference subtended by the same arc?
Question 2
In a cyclic quadrilateral, one angle is 110 degrees. What is the opposite angle?
Question 3
A line from the centre of a circle meets a tangent at the point of tangency. The angle between them is:
Key Concepts Summary
- ●The angle at the centre is twice the angle at the circumference subtended by the same arc
- ●Angles in the same segment are equal
- ●The angle in a semicircle is always 90 degrees
- ●Opposite angles in a cyclic quadrilateral sum to 180 degrees
- ●The tangent to a circle is perpendicular to the radius at the point of contact