4.5.2

Circle Theorems

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Circle Theorems

Perpendicular bisector goes through centre

Perpendicular bisector goes through centre

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

Proof

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

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

Proof

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.
Proof continued

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

Proof

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.
Proof continued

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.
Alternative proof

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
1

Numbers

1.1

Integers

1.1.1Integers1.1.2Ordering Numbers1.1.3The Four Rules of Operations1.1.4Hierarchy of Operations1.1.5Definitions & Terms1.1.6HCF & LCM1.1.7Standard Form1.1.8Standard Form 21.1.9Prime Numbers1.1.10End of Topic Test - Standard Form
1.2

Fractions

1.2.1Fractions1.2.2Using Fractions1.2.3Conversion1.2.4Reciprocals
1.3

Decimals

1.3.1Decimals1.3.2Decimals 21.3.3Exam-Style Questions - Recurring Decimals1.3.4Comparing & Converting Decimals1.3.5End of Topic Test - Fractions & Decimals
1.4

Powers & Roots

1.4.1Square Numbers1.4.2Index Notation1.4.3Exam-Style Questions - Index Notation1.4.4Advanced Indices1.4.5Surds1.4.6End of Topic Test - Powers & Roots
1.5

Set Language & Notation

1.5.1Sets
1.6

Percentages

1.6.1Percentages1.6.2Percentage Change1.6.3Percentages in Finance1.6.4Compound Interest
1.7

Ratio & Proportion

1.7.1Manipulating Ratios1.7.2Proportion
1.8

Degree of Accuracy

1.8.1Rounding & Significant Figures1.8.2Estimation1.8.3Upper & Lower Bounds1.8.4Limits of Accuracy1.8.5Limits of Accuracy 21.8.6End of Topic Test - Rounding & Estimation
2

Equations, Formulae & Identities

2.1

Algebraic Manipulation

2.1.1Algebra Basics2.1.2Power Rules & Collecting Terms2.1.3Expanding Brackets2.1.4Expanding Brackets 22.1.5Factorising Quadratics2.1.6Algebraic Fractions2.1.7Proofs2.1.8End of Topic Test - Algebra & Proofs2.1.9Exam-Style Questions - Algebraic Proof
2.2

Expressions & Formulae

2.2.1Rearranging Formulae2.2.2Simple Formulae2.2.3End of Topic Test - Algebra Manipulation & Formula
2.3

Linear Equations

2.3.1Solving Linear Equations2.3.2Simultaneous Linear Equations
2.4

Quadratic Equations

2.4.1Solving Quadratics2.4.2Solving Quadratics 22.4.3End of Topic Test - Solving Equations2.4.4Simulateous Quadratic Equations2.4.5Exam-Style Questions - Simultaneous Equations2.4.6Graphs of Simultaneous Equations2.4.7End of Topic Test - Simultaneous Equations
2.5

Proportion

2.5.1Graphs & Proportion
2.6

Inequalities

2.6.1Solving Inequalities2.6.2Graphs of Inequalities2.6.3Exam-Style Questions - Inequalities2.6.4End of Topic Test - Inequalities
3

Sequences, Functions & Graphs

3.1

Sequences

3.1.1Sequence Rules3.1.2Finding Terms3.1.3Exam-Style Questions - Sequences3.1.4End of Topic Test - Sequences
3.2

Functions

3.2.1Functions3.2.2Composite & Inverse3.2.3End of Topic Test - Functions
3.3

Graphs

3.3.1Plotting Graphs3.3.2Straight Line Graphs3.3.3Properties of Graphs3.3.4Quadratic Graphs3.3.5Practical Graphs3.3.6Graphs of Polynomials3.3.7End of Topic Test Graphs
3.4

Common Graphs

3.4.1Graphs of Common Functions3.4.2Transformations of Graphs3.4.3End of Topic Test - Transformations and Graph
3.5

Calculus

3.5.1Derivatives of Functions3.5.2Maxima & Minima
4

Geometry

4.1

Angles, Lines & Triangles

4.1.1Angles & Lines4.1.2Angle Rules4.1.3Angles in Triangles4.1.4Exam-Style Questions - Angles
4.2

Polygons

4.2.1Polygons4.2.2Quadrilaterals4.2.3Congruence4.2.4Scale Factor4.2.5Exam-Style Questions - Enlargements4.2.6End of Topic Test - Geometry
4.3

Symmetry

4.3.1Symmetry4.3.2Exam-Style Questions - Reflection
4.4

Measures

4.4.1Measurements4.4.2Time4.4.3Bearings4.4.4Speed
4.5

Circle Properties

4.5.1Properties of a Circle4.5.2Circle Theorems4.5.3Circle Theorems 24.5.4Circle Theorems 34.5.5Exam-Style Questions - Circle Theorems
4.6

Trigonometry & Pythagoras’ theorem

4.6.1Pythagoras' Theorem4.6.2Trigonometry4.6.3Sine & Cosine for Obtuse Angles4.6.4Triangle Formulae4.6.5Exam-Style Questions - Trigonometry & Area
4.7

Mensuration of 2D Shapes

4.7.1Standard Units4.7.2Perimeter of 2D Shapes4.7.3Area of 2D Shapes4.7.4Area & Circumference of Circle4.7.5End of Topic Test - Trigonometry & Shapes
4.8

3D Shapes & Volume

4.8.13D Shapes4.8.2Surface Area4.8.3Volume4.8.4Exam-Style Questions - Volume
5

Vectors & Transformation Geometry

5.1

Vectors

5.1.1Vectors5.1.2Vectors 25.1.3Exam-Style Questions - Vectors
5.2

Transformation Geometry

5.2.1Transformations5.2.2Rotations5.2.3Translations & Reflections5.2.4Enlargements5.2.5Effects of Transformations
6

Statistics & Probability

6.1

Statistical Measures

6.1.1Basic Statistics6.1.2Frequency Table
6.2

Graphical Representation of Data

6.2.1Pictograms, Bar Charts & Pie Charts6.2.2Histograms6.2.3Cumulative Frequency6.2.4Cumulative Frequency 26.2.5Exam-Style Questions - Cumulative Frequency6.2.6End of Topic - Datasets
6.3

Probability

6.3.1Probability6.3.2Theoretical Probability6.3.3Experimental Probability6.3.4Venn Diagrams6.3.5Sample Space Diagrams6.3.6Tree Diagrams6.3.7Independent Events6.3.8Conditional Probability6.3.9End of Topic Test - Probability6.3.10Exam-Style Questions - Probability

Practice questions on Circle Theorems

Can you answer these? Test yourself with free interactive practice on Seneca — used by over 10 million students.

  1. 1
    The blue line is a radius of the circle, the red line is a chord. The blue line splits the red line equally, and the two lines intersect at an angle x. What is x?Multiple choice
  2. 2
    The blue line is a radius of the circle, the red line is a chord. The blue line splits the red line equally, and the two lines intersect at an angle x. What is x?Multiple choice
  3. 3
    Which two lines are the same length?Multiple choice
  4. 4
    What is the value of angle x?Multiple choice
  5. 5
    What is the value of angle x?Multiple choice
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Properties of a Circle

Circle Theorems 2