GCSE Higher Ch16 Circle Theorems

Nicola Trubridge, Sep 20, 2017

Table of Contents

  1. Specification Details
    1. Circle Theorems
    2. Isosceles Triangle in a Circle (page 1)
  2. Theorem 1
    1. Angle at centre is twice angle at circumference
    2. Angle at centre is twice angle at circumference.
  3. Theorem 2
    1. Angle in a semi-circle
    2. Angle in a semi-circle
    3. Angle in a semi-circle PROOF
    4. Angle in a semi-circle (proof)
  4. Theorem 3
    1. Angles in the same Segment
    2. Angles in the same Segment (Proof)
  5. Theorem 4
    1. Cyclic quadrilaterals
    2. Simple Cyclic Quadrilateral
    3. Cyclic Quadrilateral (Proof)
  6. Theorem 5
    1. Tangent at a Point
  7. Theorem 6
    1. Tangents to a circle are equal (Proof)
  8. Theorem 7
    1. Diameter-Chord Theorem
    2. Diameters & Chords I (VA)
  9. Theorem 8
    1. Alternate Segment Theorem (page 1)
    2. Alternate Segment Theorem (page 2)
    3. Alternate Segment Theorem (proof)
  10. Mixed Examples
    1. Circle theorems

1.

Specification Details

  • 1. Circle Theorems
  • 2. Isosceles Triangle in a Circle (page 1)

1.1.

Circle Theorems

AQA Specification Reference G10
[b]Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results[/b][br][br]1. Angle subtended by an arc at the centre is equal to twice the angle subtended at any point on the circumference[br][br]2. Angle subtended at the circumference by a semicircle is 90°[br][br]3. Angles in the same segment are equal[br][br]4. Opposite angles in a cyclic quadrilateral sum to 180°, [br][br]5. Tangent at any point on a circle is perpendicular to the radius at that point[br][br]6. Tangents from an external point are equal in length,[br][br]7. The perpendicular from the centre to a chord bisects the chord, [br][br]8. Alternate segment theorem.
– Nicola Trubridge

1.2.

–

2.

Theorem 1

  • 1. Angle at centre is twice angle at circumference
  • 2. Angle at centre is twice angle at circumference.

2.1.

Angle at centre is twice angle at circumference

Angle subtended by an arc at the centre is equal to twice the angle subtended at any point on the circumference
– Nicola Trubridge

2.2.

–

3.

Theorem 2

  • 1. Angle in a semi-circle
  • 2. Angle in a semi-circle
  • 3. Angle in a semi-circle PROOF
  • 4. Angle in a semi-circle (proof)

3.1.

Angle in a semi-circle
– shane.williams

3.2.

–

3.3.

–

3.4.

–

4.

Theorem 3

  • 1. Angles in the same Segment
  • 2. Angles in the same Segment (Proof)

4.1.

Angles in the same Segment

Angles in the same Segment
– Michael Borcherds

4.2.

–

5.

Theorem 4

  • 1. Cyclic quadrilaterals
  • 2. Simple Cyclic Quadrilateral
  • 3. Cyclic Quadrilateral (Proof)

5.1.

Cyclic quadrilaterals

– Nicola Trubridge, Michael Borcherds

5.2.

–

5.3.

–

6.

Theorem 5

  • 1. Tangent at a Point

6.1.

Tangent at a Point

5. Tangent at any point on a circle is perpendicular to the radius at that point
– Nicola Trubridge, Broadrick Secondary School

7.

Theorem 6

  • 1. Tangents to a circle are equal (Proof)

7.1.

Tangents to a circle are equal (Proof)

Tangents from an external point are equal in length
– Nicola Trubridge, Michael Borcherds

8.

Theorem 7

  • 1. Diameter-Chord Theorem
  • 2. Diameters & Chords I (VA)

8.1.

Diameter-Chord Theorem

7. The perpendicular from the centre to a chord bisects the chord,
– Nicola Trubridge, amit.quackenbush

8.2.

–

9.

Theorem 8

  • 1. Alternate Segment Theorem (page 1)
  • 2. Alternate Segment Theorem (page 2)
  • 3. Alternate Segment Theorem (proof)

9.1.

Alternate Segment Theorem (page 1)

Alternate Segment Theorem (page 1)
– Michael Borcherds

9.2.

–

9.3.

–

10.

Mixed Examples

  • 1. Circle theorems

10.1.

Circle theorems

Demonstration of circle theorems
– Catherine Berry

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