Understanding Circle Theorems and Angles

Sep 25, 2024

Circle Theorems

Introduction

  • Understanding Circle Theorems: Finding angles within circles containing other shapes.
  • Shapes are usually not drawn to scale, adding complexity.

Key Concepts

1. Chords

  • Definition: A line below the diameter connecting two points on the circle.
  • Perpendicular bisector of a chord passes through the center of the circle.

2. Semicircles and Angles

  • A semicircle can contain a triangle.
  • Angle B in a triangle inscribed in a semicircle is always 90°.
  • This holds true for any triangle inscribed in a semicircle.

3. Tangents

  • Definition: A line that touches the circle at one point.
  • The angle formed between a radius and a tangent at the point of contact is 90°.

4. Subtended Angles

  • Concept: The angle formed by lines from the ends of an arc to a point on the circle.
  • Example: If a tree and a person create a subtended angle, it can be measured.

5. Angles in the Same Segment

  • Angles subtended by the same arc are equal.
  • Example: Angle ABC is the same as angle ADC.

6. Angles in an Arrowhead Shape

  • The angle inside is twice the angle at the circumference.

7. Quadrilaterals in Circles

  • In a cyclic quadrilateral (four-sided figure), the opposite angles sum to 180°.
  • Example: Angles A and C sum to 180°, same for angles B and D.

8. Lengths of Tangents

  • The lengths of two tangents from a single point outside the circle to the circle are equal.
  • Example: Length AB = Length AC.

9. Angle Between Tangent and Chord

  • The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment (Alternate Segment Theorem).

Example Problem

  • Given: Angle A = 70°.
  • Calculation steps:
    1. Angle A + Angle B = 180° => Angle B = 110°.
    2. Divide by 2 to find angles in isosceles triangle => 55°.
    3. Use properties of the tangent to find remaining angles.
  • Final angle D found to be 55°.

Conclusion

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