Circle Theorem Questions Lecture Notes

Introduction

• Lecture Focus: Solving circle theorem problems.
• Structure: Covering five easier questions first, then three exam-specific questions at the end.

Question 1: Angle D, O, A

• Given:
  • BO to D is the diameter.
  • BC and AC are tangents.
  • Find angle D, O, A (labeled as X).
• Steps:
  • Utilize tangents and radii to identify 90-degree angles.
  • Identify the triangle involved and use 180-degree rule.
  • Using two tangents forming equal lengths, which bisect angles.
  • Add angles and subtract from 180 for the straight line.
• Final Answer: 68 degrees.
• Key Concepts:
  • Tangents meet radius at 90 degrees.
  • Isosceles triangles formed by tangents.
  • Angles on a straight line.

Question 2: Angle AEC

• Given:
  • Angle ADC is needed.
  • Given angle at the center from A to C through O.
  • Circle forms a quadrilateral with two other marked angles.
• Steps:
  • Find angle at circumference (half of the central angle given).
  • Use cyclic quadrilateral properties (opposite angles sum to 180).
• Final Answer: 96 degrees.
• Key Concepts:
  • Angles at the center and circumference.
  • Cyclic quadrilateral angle properties.

Question 3: Angle BP tangent

• Given:
  • AP and BP are tangents.
  • Isosceles triangle formed by radii.
  • Angle X provided and 86 degrees in outer angle.
• Steps:
  • Find base angles using isosceles properties.
  • Deduct each angle from their respective 90-degree points.
  • Determine X using other known angles in the isosceles triangles.
• Final Answer: 43 degrees.
• Key Concepts:
  • Tangent meets radius at 90 degrees.
  • Isosceles triangle properties.

Question 4: Tangent with ABC and ADE angles

• Given:
  • Tangent ABC.
  • Diameter BIE.
• Steps:
  • Utilize tangent meeting radius rules for initial angles.
  • Use isosceles triangle properties for follow-up angles.
• Final Answer: 55 degrees and 55 degrees using alternate segment theorem.
• Key Concepts:
  • Tangents and radii forming 90 degrees.
  • Alternate segment theorem.

Question 5: Tangent to Circle, Angle OTQ

• Given:
  • ATB tangent.
  • Visible isosceles triangle with tangents.
• Steps:
  • Find 90-degree points and deduct given angles.
  • Apply alternate segment theorem.
  • Use triangle base angles to find missing angle.
• Final Answer: 29 degrees.
• Key Concepts:
  • Tangent and radius.
  • Alternate segment theorem.
  • Isosceles triangle.

Exam-Style Questions

Question 1: AOB Angle

• Given: Tangent and radius intersection determining initial angles.
• Steps:
  • Apply radius-tangent 90-degree rule.
  • Use isosceles and cyclic quadrilateral properties to compute other angles.
• Final Answer: 66 degrees
• Key Concepts: Isosceles triangle and cyclic quadrilateral.

Question 2: CAD Angle

• Given: ABC angle.
• Steps:
  • Use cyclic quadrilateral rules.
  • Use the tangent and isosceles triangle rules to obtain further insights.
  • Apply alternate segment theorem to finalize the angle.
• Final Answer: 60 degrees.
• Key Concepts: Tangent and radius, cyclic quadrilateral, alternate segment theorem

Question 3: CAO Angle

• Given: apllies radius and tangent intersection for initial derivation.
• Steps:
  • Utilize isosceles triangle properties step by step to reduce complexity.
• Final Answer: 21 degrees.
• Key Concepts: Tangent and radius, isosceles triangle properties.

Conclusion

• Upcoming Topics: Algebra within circle theorems, harder circle theorem questions.
• Recommendations: Review marked points, re-attempt unsolved problems.