Lecture Notes: Theorems with Circles

Introduction

• Focus on theorems related to circles.
• Previously covered:
  • Lengths of arcs
  • Areas of circle segments
  • Circumference and area of a circle
• Aim: Understand formulas and concepts, not just memorize them.

Key Circle Components

Definitions

• Diameter: Line through the center, divides circle into two equal halves.
• Radius: Line from center to any point on the circle.
• Chord: Line connecting two points on a circle.
• Secant: Extended chord, intersects circle at two points.
• Tangent: Line touching the circle at exactly one point.
• Arc: Portion of the circle's circumference.
  • Minor Arc: Less than 180°.
  • Major Arc: More than 180°.

Tangent Theorem

• Tangent Theorem: A line is tangent to a circle if and only if it is perpendicular to the radius at the point of tangency.
  • If and only if: Works both ways.
  • 90° Angle: Tangent creates a right angle with the radius.

Practice Problems

• Pythagorean Theorem: Apply with right triangles formed by radius and tangent.
  • Example:
    • Given R and a tangent segment, use Pythagorean theorem to find unknown lengths.

Tangent Segment Theorem

• Theorem: Tangent segments from a common external point are congruent.
• Application: Set equations equal to one another to solve for unknowns.

Central and Inscribed Angles

Definitions

• Central Angle: Vertex at circle's center, endpoints on circumference.
• Inscribed Angle: All points on the circle.

Arc Measurement

• Arc Degree Measure: Same as central angle degree.
• Central vs Inscribed Angle: Central angle is twice the inscribed angle.

Practice: Angle Comparisons

• Equal Inscribed Angles: Inscribed angles subtending the same arc are equal.

Inscribed Angle Theorem

• Right Triangle Theorem: Inscribed angle on a diameter results in a 90° angle.
  • Applications: Use in problems involving right triangles and circle diameters.

Quadrilateral Inscribed Theorem

• Theorem: Opposite angles of an inscribed quadrilateral are supplementary (add up to 180°).

Conclusion

• Review these theorems and practice applying them to problems.
• Ensure understanding not just memorization.
• Preparation for continued study of circle theorems in future lessons.