Solving Circle Theorems - lesson 72

Sep 27, 2024

Circle Theorems Part 4

Introduction

  • Focus: Hints and tips for tackling exam questions using circle theorems.
  • Key Point: Multiple circle theorems may be used in a single question.

Question 1: Points A, B, C, D on Circle

  • Given:
    • A, B, C, D are points on the circle's circumference, center O.
    • AC is a diameter, AC and BD intersect at E.
    • Angle CAB = 25 degrees, DEC = 100 degrees.
  • Task: Find angle DAC.
  • Solution Steps:
    1. Use Diameter:
      • Angle ABC = 90 degrees (Angle subtended by semicircle).
    2. Calculate ACB:
      • 180 - 90 - 25 = 65 degrees (Angles in a triangle sum to 180 degrees).
    3. Use Arc Theorem:
      • Angle ACB = Angle ADB = 65 degrees (Angles subtended by same arc).
    4. Calculate AED:
      • 180 - 100 = 80 degrees (Angles on a straight line).
    5. Find DAC:
      • 180 - 80 - 65 = 35 degrees (Angles in a triangle).

Question 2: Points B, D, E, F on Circle

  • Given:
    • B, D, E, F are on the circle.
    • ABC is the tangent at B.
  • Task: Find angle ABD.
  • Solution Steps:
    1. Use Cyclic Quadrilateral:
      • Angle DBF = 180 - 100 = 80 degrees (Opposite angles sum to 180).
    2. Calculate BFD:
      • 180 - 80 - 40 = 60 degrees (Angles in a triangle).
    3. Use Alternate Segment Theorem:
      • Angle ABD = 60 degrees (Equal to angle BFD).

Question 3: Points B, C, D on Circle

  • Given:
    • ABE and ADF are tangents.
    • Angle DAB = 40 degrees, CBE = 75 degrees.
  • Task: Find angle ODC.
  • Solution Steps:
    1. Use Tangent-Radius Property:
      • Angles ADO and ABO = 90 degrees.
    2. Calculate DOB:
      • 360 - 90 - 90 - 40 = 140 degrees (Angles in a quadrilateral).
    3. Use Center-Circumference Theorem:
      • Angle DCB = 70 degrees (140/2).
    4. Calculate OBC:
      • 90 - 75 = 15 degrees.
    5. Find Reflex DOB:
      • 360 - 140 = 220 degrees.
    6. Calculate ODC:
      • 360 - 220 - 70 - 15 = 55 degrees.

Question 4: Points A, B, C, D on Circle

  • Task: Show Y - X = 90 degrees.
  • Solution Steps:
    1. Use Cyclic Quadrilateral:
      • Angle BCD = 180 - Y.
    2. Use Center-Circumference Theorem:
      • 2(100 - Y) = 360 - 2Y.
    3. Isosceles Triangle Property:
      • OBD = X.
    4. Set Equation:
      • 180 = X + X + (360 - 2Y).
    5. Solve for Y - X:
      • 180 = 2Y - 2X, thus 90 = Y - X.

Summary

  • Memorize circle theorems and key angle facts.
  • Extract key words from questions to identify applicable theorems.
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