Circle theorems - Part 1

Circle Theorems - Part 1

Theorem One: Angle Subtended by Diameter

• The angle subtended by a diameter of a circle at the circumference is always a right angle (90 degrees).
• This means if you have a semi-circle, any triangle formed with the end points of the diameter and a point on the circumference will be a right-angle triangle.

Theorem Two: Angles Subtended by Same Arc

• If two or more angles subtend the same arc (or, equivalently, are created by the same ‘slices’ of the circle), they are equal, regardless of their position on the circumference.
• For instance, if two angles are both subtended by the same two points on the circumference, they are equal, regardless of where they are positioned in relation to the centre of the circle.

Theorem Three: Angle at the Centre

• The angle subtended by an arc at the centre of the circle is twice the angle subtended by the same arc at the circumference.
• This means if you have an arc of a circle, and you draw lines from the end points of the arc to both the circumference and the centre of the circle, the angle at the centre will be twice that at the circumference.

Theorem Four: Cyclic Quadrilateral

• In a cyclic quadrilateral, the opposite angles add up to 180 degrees.
• A cyclic quadrilateral is a four-sided figure where all corners (vertices) lie on the circumference of the same circle.

Theorem Five: Tangent and Radius

• A tangent to a circle forms a right angle with the radius drawn to the point of tangency.
• So if a line just touches a circle at one point and you draw a line from the centre of the circle to the point where the circle and line touch, a right-angle triangle will be formed.

Remember to memorise these theorems and apply them to solve geometry problems involving circles.