4.5.2

Circle Theorems

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Circle Theorems

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Perpendicular bisector goes through centre

  • The perpendicular bisector of a chord always goes through the centre of the circle.
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Proof

  • Draw the perpendicular from centre, O, to the chord AB.
  • Draw a radius to A and another to B to form two triangles.
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Proof continued

  • Their hypotenuses are the same length (because they are both radii) and they share another edge so the triangles are congruent by the ‘RHS’ rule.
  • Therefore AM = BM and so the perpendicular splits the chord exactly in half.

Circle Theorems

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Proof

  • Split the shape into two isosceles triangles and label the angles.
  • The angle at the centre is 360° - x - y and the angle at the circumference is a + b.
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Proof continued

  • Since the angles in a triangle add up to 180°:
    a = 1⁄2(180° - y), b = 1⁄2(180° - x)
  • So angle at circumference = a + b = 1⁄2(360° - x - y)
    = 1⁄2 × angle at centre.

Circle Theorems

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Proof

  • Split the triangle into two triangles which are both isosceles since they both have two sides which are radii.
  • Mark one of the angles at the centre x.
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Proof continued

  • y = 1⁄2(180° - x) since the triangle is isosceles and all angles add up to 180°.
  • Similarly z = 1⁄2(180° - (180° - x)) = 1⁄2x
  • Therefore the angle at the circumference is z + y = 1⁄2 × 180° = 90° as required.
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Alternative proof

  • Alternatively, using the previous theorem we see that the angle at the centre is twice the angle at the circumference.
  • So 180° is twice the angle at the circumference so the angle is 90°.

Jump to other topics

1Numbers

1.1Integers

1.1.1Integers

1.1.2Ordering Numbers

1.1.3The Four Rules of Operations

1.1.4Hierarchy of Operations

1.1.5Definitions & Terms

1.1.6HCF & LCM

1.1.7Standard Form

1.1.8Standard Form 2

1.1.9Prime Numbers

1.1.10End of Topic Test - Standard Form

1.2Fractions

1.2.1Fractions

1.2.2Using Fractions

1.2.3Conversion

1.2.4Reciprocals

1.3Decimals

1.3.1Decimals

1.3.2Decimals 2

1.3.3Exam-Style Questions - Recurring Decimals

1.3.4Comparing & Converting Decimals

1.3.5End of Topic Test - Fractions & Decimals

1.4Powers & Roots

1.4.1Square Numbers

1.4.2Index Notation

1.4.3Exam-Style Questions - Index Notation

1.4.4Advanced Indices

1.4.5Surds

1.4.6End of Topic Test - Powers & Roots

1.5Set Language & Notation

1.5.1Sets

1.6Percentages

1.6.1Percentages

1.6.2Percentage Change

1.6.3Percentages in Finance

1.6.4Compound Interest

1.7Ratio & Proportion

1.7.1Manipulating Ratios

1.7.2Proportion

1.8Degree of Accuracy

1.8.1Rounding & Significant Figures

1.8.2Estimation

1.8.3Upper & Lower Bounds

1.8.4Limits of Accuracy

1.8.5Limits of Accuracy 2

1.8.6End of Topic Test - Rounding & Estimation

2Equations, Formulae & Identities

2.1Algebraic Manipulation

2.1.1Algebra Basics

2.1.2Power Rules & Collecting Terms

2.1.3Expanding Brackets

2.1.4Expanding Brackets 2

2.1.5Factorising Quadratics

2.1.6Algebraic Fractions

2.1.7Proofs

2.1.8End of Topic Test - Algebra & Proofs

2.1.9Exam-Style Questions - Algebraic Proof

2.2Expressions & Formulae

2.2.1Rearranging Formulae

2.2.2Simple Formulae

2.2.3End of Topic Test - Algebra Manipulation & Formula

2.3Linear Equations

2.3.1Solving Linear Equations

2.3.2Simultaneous Linear Equations

2.4Quadratic Equations

2.4.1Solving Quadratics

2.4.2Solving Quadratics 2

2.4.3End of Topic Test - Solving Equations

2.4.4Simulateous Quadratic Equations

2.4.5Exam-Style Questions - Simultaneous Equations

2.4.6Graphs of Simultaneous Equations

2.4.7End of Topic Test - Simultaneous Equations

2.5Proportion

2.5.1Graphs & Proportion

2.6Inequalities

2.6.1Solving Inequalities

2.6.2Graphs of Inequalities

2.6.3Exam-Style Questions - Inequalities

2.6.4End of Topic Test - Inequalities

3Sequences, Functions & Graphs

3.1Sequences

3.1.1Sequence Rules

3.1.2Finding Terms

3.1.3Exam-Style Questions - Sequences

3.1.4End of Topic Test - Sequences

3.2Functions

3.2.1Functions

3.2.2Composite & Inverse

3.2.3End of Topic Test - Functions

3.3Graphs

3.3.1Plotting Graphs

3.3.2Straight Line Graphs

3.3.3Properties of Graphs

3.3.4Quadratic Graphs

3.3.5Practical Graphs

3.3.6Graphs of Polynomials

3.3.7End of Topic Test Graphs

3.4Common Graphs

3.4.1Graphs of Common Functions

3.4.2Transformations of Graphs

3.4.3End of Topic Test - Transformations and Graph

3.5Calculus

3.5.1Derivatives of Functions

3.5.2Maxima & Minima

4Geometry

4.1Angles, Lines & Triangles

4.1.1Angles & Lines

4.1.2Angle Rules

4.1.3Angles in Triangles

4.1.4Exam-Style Questions - Angles

4.2Polygons

4.2.1Polygons

4.2.2Quadrilaterals

4.2.3Congruence

4.2.4Scale Factor

4.2.5Exam-Style Questions - Enlargements

4.2.6End of Topic Test - Geometry

4.3Symmetry

4.3.1Symmetry

4.3.2Exam-Style Questions - Reflection

4.4Measures

4.4.1Measurements

4.4.2Time

4.4.3Bearings

4.4.4Speed

4.5Circle Properties

4.5.1Properties of a Circle

4.5.2Circle Theorems

4.5.3Circle Theorems 2

4.5.4Circle Theorems 3

4.5.5Exam-Style Questions - Circle Theorems

4.6Trigonometry & Pythagoras’ theorem

4.6.1Pythagoras' Theorem

4.6.2Trigonometry

4.6.3Sine & Cosine for Obtuse Angles

4.6.4Triangle Formulae

4.6.5Exam-Style Questions - Trigonometry & Area

4.7Mensuration of 2D Shapes

4.7.1Standard Units

4.7.2Perimeter of 2D Shapes

4.7.3Area of 2D Shapes

4.7.4Area & Circumference of Circle

4.7.5End of Topic Test - Trigonometry & Shapes

4.83D Shapes & Volume

4.8.13D Shapes

4.8.2Surface Area

4.8.3Volume

4.8.4Exam-Style Questions - Volume

5Vectors & Transformation Geometry

5.1Vectors

5.1.1Vectors

5.1.2Vectors 2

5.1.3Exam-Style Questions - Vectors

5.2Transformation Geometry

5.2.1Transformations

5.2.2Rotations

5.2.3Translations & Reflections

5.2.4Enlargements

5.2.5Effects of Transformations

6Statistics & Probability

6.1Statistical Measures

6.1.1Basic Statistics

6.1.2Frequency Table

6.2Graphical Representation of Data

6.2.1Pictograms, Bar Charts & Pie Charts

6.2.2Histograms

6.2.3Cumulative Frequency

6.2.4Cumulative Frequency 2

6.2.5Exam-Style Questions - Cumulative Frequency

6.2.6End of Topic - Datasets

6.3Probability

6.3.1Probability

6.3.2Theoretical Probability

6.3.3Experimental Probability

6.3.4Venn Diagrams

6.3.5Sample Space Diagrams

6.3.6Tree Diagrams

6.3.7Independent Events

6.3.8Conditional Probability

6.3.9End of Topic Test - Probability

6.3.10Exam-Style Questions - Probability

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Properties of a Circle

Circle Theorems 2