Circle Theorems Lecture Notes

Key Terms in Circle Geometry

• Chord: A straight line from one side of the circle to the other, splitting it into two segments.
• Segment: The areas of the circle divided by a chord.

Circle Theorems

1. Angles in the Same Segment

• Theorem: Angles in the same segment are equal.
• Application: If multiple angles are created using the same chord in one segment, they are all equal.
• Visualization: Often looks like a bow tie.
• Example: If one angle is 68 degrees, all angles in the same segment are 68 degrees.

2. Angle in a Semicircle

• Theorem: The angle formed in a semicircle is always a right angle (90 degrees).
• Application: Use a diameter as the chord for these angles.

3. Angle at the Center vs. Angle at the Circumference

• Theorem: The angle at the center is twice the angle at the circumference.

4. Cyclic Quadrilaterals

• Definition: A quadrilateral where all four vertices touch the circle’s circumference.
• Theorem: Opposite angles of a cyclic quadrilateral sum to 180 degrees.

5. Tangent and Radius

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

6. Tangents from a Point

• Theorem: Tangents drawn from a single external point to a circle are equal in length.

7. Alternate Segment Theorem

• Theorem: The angle between the tangent and the chord through the point of contact is equal to the angle in the alternate segment.

Problem-Solving with Circle Theorems

Problem 1: Finding Angle ABD

• Steps:
  • Use angles on a straight line to determine unknown angles.
  • Apply the theorem of angles in the same segment to find equal angles.

Problem 2: Finding Angle BCF

• Steps:
  • Use properties of isosceles triangles to find angles.
  • Apply the theorem of cyclic quadrilaterals for opposite angles.
  • Use knowledge of diameter forming right angles in a semicircle.
  • Apply alternate segment theorem to find congruent angles.

Problem 3: Complex Circle Geometry

• Steps:
  • Use tangent-radius theorem for right angles.
  • Calculate unknown angles in triangles by sum of angles.
  • Apply center-circumference theorem for angle relationships.
  • Identify isosceles triangles from radii.
  • Apply sum of angles in triangles for final calculation.

Conclusion

• Reference diagram showing all theorems for revision.
• Emphasize different methods to solve problems, stressing clear explanation and methodology for full marks.