Circle Theorems Lecture Notes

Key Terms and Definitions

• Chord: A straight line going from one side of a circle to the other.
• Segment: A chord splits a circle into two segments (top and bottom).
• Diameter: A chord passing through the center of a circle.
• Cyclic Quadrilateral: A quadrilateral with all vertices on the circumference of the circle.
• Tangent: A straight line that touches a circle at one point.
• Radius: A line from the center of the circle to any point on its circumference.

Theorems

Theorem 1: Angles in the Same Segment

• Statement: Angles in the same segment are equal.
• Illustration: When angles are created using the same chord and within the same segment, they are equal.
• Example: If three angles are created by the same chord in one segment, all angles are equal.

Theorem 2: Angle in a Semicircle

• Statement: The angle in a semicircle is 90 degrees.
• Illustration: When a diameter is used as a chord, all angles formed in the semicircle are right angles.

Theorem 3: Angle at the Center vs. Circumference

• Statement: The angle at the center is twice the angle at the circumference.
• Example: If the angle at the circumference is 50 degrees, the angle at the center is 100 degrees.

Theorem 4: Cyclic Quadrilateral

• Statement: Opposite angles in a cyclic quadrilateral add up to 180 degrees.
• Example: If one angle is 86 degrees, the opposite angle is 94 degrees.

Theorem 5: Tangent and Radius

• Statement: A tangent meets a radius at 90 degrees.

Theorem 6: Tangents from a Point

• Statement: Tangents from the same point are equal in length.
• Additional: A line from the center bisects the angle between the tangents.

Theorem 7: Alternate Segment Theorem

• Statement: The angle between the tangent and chord is equal to the angle in the alternate segment.

Problem Solving

Example 1: Finding Angles

• Given: Various angles and chords in a circle.
• Process:
  • Use straight-line theorem (angles on a straight line add to 180 degrees).
  • Apply first circle theorem (angles in the same segment are equal).
• Example 2: Cyclic quadrilateral and semi-circle theorems applied to find unknown angles.

Example 2: Using Tangents and Radii

• Given: Tangent and radius form right angles.
• Process:
  • Identify isosceles triangles and apply cyclic quadrilateral theorem.
  • Use interior angles of triangle theorem to find missing angles.

Conclusion

• Review of all theorems and how they are used to solve geometrical problems related to circles.
• Emphasis on marking diagrams and showing clear working out for full credit in exams.

Suggestions: Practice with exam questions using these theorems for better understanding and application.