4.15.3

Angle of a Semicircle

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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

Number

1.1

Place Value & Ordering

1.1.1Place Value
1.2

Numerical Operations

1.2.1Adding Integers1.2.2Subtracting Integers1.2.3Multiplying Integers1.2.4Dividing Integers1.2.5Adding & Subtracting Decimals1.2.6Multiplying Decimals1.2.7Dividing Decimals1.2.8Adding & Subtracting Negative Numbers1.2.9Multiplying & Dividing Negative Numbers1.2.10Multiplying by Powers of Ten1.2.11Dividing by Powers of Ten
1.3

Structure of Calculations

1.3.1Order of Operations (BIDMAS)
1.4

Fractions

1.4.1Writing Fractions1.4.2Simplifying Fractions1.4.3Fractions & Recurring Decimals
1.5

Numerical Operations on Fractions

1.5.1Adding & Subtracting Fractions1.5.2Adding & Subtracting Mixed Numbers1.5.3Multiplying Fractions1.5.4Multiplying Mixed Numbers1.5.5Dividing Fractions1.5.6Dividing Mixed Numbers (sample)1.5.7Dividing Mixed Numbers
1.6

Roots & Powers

1.6.1Roots1.6.2Fractional Indices (1/n)1.6.3Fractional Indices (m/n)1.6.4Laws of Indices (Multiplication)1.6.5Laws of Indices (Division)
1.7

Numerical Skills

1.7.1Using a Calculator - Powers, Roots & Fractions1.7.2Using a Calculator - Trigonometric Functions
1.8

Factors & Multiples

1.8.1Finding Factors1.8.2Finding Multiples
1.9

Surds

1.9.1Multiplying & Dividing Surds1.9.2Simplifying Surds1.9.3Expanding Expressions With Surds1.9.4Rationalising the Denominator
1.10

Standard Form

1.10.1Converting to Standard Form (Positive Powers)1.10.2Converting to Standard Form (Negative Powers)1.10.3Converting From Standard Form (Positive Powers)1.10.4Converting From Standard Form (Negative Powers)1.10.5Adding & Subtracting Standard Form1.10.6Multiplying & Dividing Standard Form
1.11

Fractions, Decimals & Percentages

1.11.1Converting Fractions, Decimals & Percentages1.11.2Converting Recurring Decimals1.11.3Numbers as Percentages1.11.4Fractions of Amounts
1.12

Percentage Change

1.12.1Percentages of Amounts (No Calculator)1.12.2Percentage Change1.12.3Original Values After Percentage Change1.12.4Growth & Decay Using Multipliers
1.13

Rounding & Estimation

1.13.1Rounding Integers1.13.2Estimating Calculations1.13.3Rounding Decimals1.13.4Rounding With Significant Figures
1.14

Bounds & Accuracy

1.14.1Finding Error Intervals1.14.2Upper & Lower Bounds1.14.3Error Intervals From Truncation
1.15

Using Numbers

1.15.1Equal Symbols1.15.2Negative Numbers1.15.3BIDMAS1.15.4Inverse Operations1.15.5End of Topic Test- Ordering1.15.6Factors1.15.7Multiples1.15.8Highest Common Factor1.15.9Lowest Common Multiple1.15.10Number of Ways to Do a Task1.15.11Number of Combinations1.15.12Number of Ways To Arrange Objects1.15.13End of Topic Test - Factors, Multiples & Combos1.15.14Standard Form1.15.15Ordinary Numbers to Standard Form1.15.16Addition in Standard Form1.15.17Subtraction in Standard Form1.15.18Multiplication in Standard Form1.15.19Division in Standard Form1.15.20Exam-Style Questions - Standard Form & Primes1.15.21Long Division1.15.22Prime Numbers1.15.23End of Topic Test - Standard Form & Long Division1.15.24Diagnostic Misconceptions - 1 NOT a Prime Number1.15.25Diagnostic Misconceptions - BIDMAS1.15.26Diagnostic Misconceptions - Adding Negatives
1.16

Fractions, Decimals & Percentages

1.16.1Adding Fractions1.16.2Subtracting Fractions1.16.3Multiplying Fractions1.16.4Dividing Fractions1.16.5Mixed Fractions1.16.6Calculating Percentages of Numbers1.16.7Calculating Fractions of Numbers1.16.8Multiply Decimals1.16.9Place Value1.16.10Recurring Decimals1.16.11Exam-Style Questions - Recurring Decimals1.16.12Convert Fractions & Terminating Decimals1.16.13Convert Recurring Decimals1.16.14Convert Decimals & Percentages1.16.15Compare Decimals1.16.16Compare Fractions1.16.17Exam-Style Questions - Percentages1.16.18End of Topic Test - Fractions & Decimals1.16.19Diagnostic Misconceptions - Adding Fractions
1.17

Powers & Roots

1.17.1Powers1.17.2Roots1.17.3Indices Advanced1.17.4Estimating Powers1.17.5Estimating Roots1.17.6Multiplying Powers1.17.7Dividing Powers1.17.8Power Raised to Power1.17.9Fractions Raised to a Power1.17.10Numbers Raised to Negative Powers1.17.11Roots of Numbers1.17.12Numbers to the Power of 1 & 01.17.13Surds1.17.14Simplify Surds1.17.15Rationalise Denominator1.17.16Exam-Style Questions - Indices1.17.17End of Topic Test - Powers & Roots1.17.18Diagnostic Misconceptions - Squaring1.17.19Diagnostic Misconceptions - Simplifying Surds
1.18

Accuracy

1.18.1Decimal Rounding1.18.2Significant Figure Rounding1.18.3Estimation1.18.4Truncation1.18.5Limits of Accuracy1.18.6Bounds Addition1.18.7Bounds Subtraction1.18.8Bounds Multiplication1.18.9Bounds Division1.18.10End of Topic Test - Rounding & Estimation1.18.11Diagnostic Misconceptions - Upper Bounds
2

Algebra

2.1

Algebraic Notation

2.1.1Using Algebraic Notation
2.2

Substitution & Simplification

2.2.1Substituting into Expressions2.2.2Substituting into Real-Life Formulae2.2.3Simplifying Expressions by Collecting Like Terms
2.3

Brackets

2.3.1Expanding Single Brackets2.3.2Expanding Double Brackets2.3.3Factorising into One Bracket2.3.4Factorising Quadratics of the Form x^2 + bx + c2.3.5Factorising Quadratics - AC Method2.3.6Factorising Quadratics - Middle Term Method2.3.7Factorising the Difference of Two Squares2.3.8Completing the Square
2.4

Algebraic Fractions

2.4.1Simplifying Algebraic Fractions by Cancelling2.4.2Simplifying Algebraic Fractions by Factorising2.4.3Operations with Algebraic Fractions
2.5

Rearranging

2.5.1Changing the Subject2.5.2Changing the Subject (Repeated Subject)
2.6

Functions

2.6.1Finding the Input of a Function2.6.2Inverse Functions
2.7

Solving Linear Equations

2.7.1Solving Linear Equations (One Step)2.7.2Solving Linear Equations (Multi-Step)2.7.3Linear Equations (Unknown Both Sides)2.7.4Solving Equations: (Unknown Denominator)2.7.5Forming & Solving Linear Equations
2.8

Inequalities

2.8.1Solving Linear Inequalities2.8.2Linear Inequalities with Unknowns on Both Sides
2.9

Solving Quadratic Equations

2.9.1Solving Quadratics by Factorising2.9.2Solving Quadratics Using the Formula
2.10

Simultaneous Equations

2.10.1Solving Simultaneous Equations (Elimination)
2.11

Iteration

2.11.1Iterative Formulae
2.12

Sequences

2.12.1Term-to-Term Rules2.12.2Position-to-Term Rules2.12.3Quadratic Sequence Formulae
2.13

Linear Graphs

2.13.1Equations of Straight Lines2.13.2Finding Line Equation from Two Points2.13.3Parallel Line Equations2.13.4Perpendicular Line Equations
2.14

Introduction to Algebra

2.14.1Algebra Basics2.14.2Substituting Values2.14.3Algebraic Vocabulary2.14.4Collecting Terms2.14.5End of Topic Test - Algebra Basics & Power Rules2.14.6Expanding Brackets I2.14.7Common Factors2.14.8Expanding Brackets II2.14.9Rearrange For x2.14.10Rearrange with Powers2.14.11Exam-Style Questions - Basics of Algebra2.14.12End of Topic Test - Expanding & Rearranging
2.15

Manipulating Algebra

2.15.1How to Factorise2.15.2Factorising Quadratics2.15.3Algebraic Fractions - Addition & Subtraction2.15.4Algebraic Fractions - Multiplication & Division2.15.5Diagnostic Misconceptions - Common Denominators2.15.6Factorising Quadratics (tt)2.15.7Simplify algebraic fractions - cancelling factors2.15.8Simplify algebraic fractions - factorisation
2.16

Proofs & Functions

2.16.1Identities & Proofs2.16.2Exam-Style Questions - Algebraic Proof2.16.3End of Topic Test - Algebra & Proofs2.16.4Functions2.16.5Combine Functions2.16.6Composite Functions2.16.7Inverse Functions2.16.8Functions (tt)
2.17

Straight Line Graphs

2.17.1Straight Line Graphs2.17.2y = mx + c2.17.3Grad + YIntercept2.17.4Exam-Style Questions - Straight Line Graphs2.17.5End of Topic Test - Functions & Straight Lines2.17.6Diagnostic Misconceptions - The x Coefficient
2.18

Common Graphs

2.18.1Shape of Quadratic Graphs2.18.2Roots of a Quadratic Graph2.18.3Completing Squares2.18.4Turning Points2.18.5Common Functions 12.18.6Common Functions 22.18.7Trigonometric functions2.18.8End of Topic Test - Common Graphs
2.19

Transformations & Tangents

2.19.1Transformations2.19.2Tangent on a Circle
2.20

Properties of Graphs

2.20.1Estimate Gradient2.20.2Estimate Area2.20.3Represent Gradient & Area2.20.4Distance-Time Graph2.20.5Velocity-Time Graph2.20.6End of Topic Test - Transformations & Graphs
2.21

Solving Equations

2.21.1Solve Linear Equations2.21.2Linear Equations with Fractions2.21.3Solve Linear Equations Graphically2.21.4Quadratic ±2.21.5Factorise Quadratics2.21.6Quadratic Formula2.21.7Complete the Square2.21.8Solve Quadratics Graphically2.21.9End of Topic Test - Solving Equations2.21.10Simultaneous Equations2.21.11Quadratic Simultaneous Equations2.21.12Simultaneous Equations - Graphs2.21.13Simultaneous Equations - Word Problems2.21.14Approximate Solutions - Number of Iterations2.21.15Approximate Solutions - Iterations to 1 DP2.21.16Approximate Solutions - Iteration Formula2.21.17Exam-Style Questions - Simultaneous Equations2.21.18End of Topic Test - Simultaneous Equations2.21.19Diagnostic Misconceptions - Solving Quadratics
2.22

Inequalities

2.22.1Solving Inequalities2.22.2Solving Double Inequalities2.22.3Solving Quadratic Inequalities2.22.4Representing Inequalities2.22.5Exam-Style Questions - Inequalities2.22.6End of Topic Test - Inequalities
2.23

Sequences

2.23.1Nth Term Rules2.23.2Recurrence Relationships2.23.3Find Nth Term in Linear Sequence2.23.4Check If a Number is In a Sequence2.23.5Exam-Style Questions - Sequences2.23.6Arithmetic & Geometric Progressions2.23.7Key Sequences2.23.8Quadratic Sequences2.23.9Find Nth Term in Quadratic Sequence2.23.10Exam-Style Questions - Geometric Series2.23.11End of Topic Test - Sequences2.23.12Nth Term Rules (tt)
3

Ratio

3.1

Unit Conversions

3.1.1Converting Units of Time3.1.2Metric Conversions3.1.3Converting Units of Area & Volume
3.2

Compound Units

3.2.1Calculating Speed3.2.2Density, Mass & Volume
3.3

Ratio Fundamentals

3.3.1Ratios as Fractions & Linear Equations3.3.2Sharing in a Ratio3.3.3Proportion Problems
3.4

Advanced Ratio and Scale

3.4.1Map Scales & Real Distances
3.5

Proportion

3.5.1Currency Conversions3.5.2Direct & Inverse Proportion
3.6

Algebraic Proportion

3.6.1Inverse Proportion Equations3.6.2Direct Proportion with Powers & Roots3.6.3Inverse Proportion with Powers & Roots
3.7

Ratios in Practice

3.7.1Converting Between Units3.7.2Formula Triangles3.7.3Scale Factors3.7.4Ratios in Shapes3.7.5Diagnostic Misconceptions - Writing Ratios
3.8

Manipulating Ratios

3.8.1Simplifying Ratios3.8.2Turn Ratios into Equations3.8.3Ratio Notation & Reduction3.8.4End of Topic Test - Units & Ratios3.8.5Part : Part Ratio3.8.6Part : Whole Ratio3.8.7Two Ratios3.8.8Differences in Ratios3.8.9Fractions in Ratios3.8.10Numbers as Fractions3.8.11Ratios to Fractions3.8.12Exam-Style Questions - Ratios3.8.13End of Topic Test - Fractions & Ratios3.8.14Diagnostic Misconceptions - Combining Ratios
3.9

Percentage & Interest

3.9.1Convert Percentages to Decimals3.9.2Convert Percentages to Fractions3.9.3Find the Percentage3.9.4More/Less Percentage3.9.5Price from Percentage Change3.9.6Calculate Percentage Change from Price3.9.7End of Topic Test - Percentages3.9.8Profit & Interest & Discount3.9.9Original Price after Discounts3.9.10Simple Interest3.9.11Compound Growth & Decay3.9.12Compound Interest3.9.13Compound Interest - After How Many Days?3.9.14End of Topic Test - Percentages in Finance
3.10

Proportion

3.10.1Proportion3.10.2Direct Proportion3.10.3Inverse Proportion3.10.4Exam-Style Questions - Proportionality
3.11

Gradient

3.11.1Gradient on a Straight Line3.11.2Direct & Inverse Proportions as Graphs3.11.3Gradient of a Curve3.11.4Positive & Negative Gradients3.11.5End of Topic Test - Proportion & Gradient
4

Geometry

4.1

Area & Perimeter

4.1.1Perimeter of Simple Shapes4.1.2Perimeter of Compound Shapes4.1.3Area of Triangles4.1.4Area of Rectangles & Parallelograms4.1.5Area of Trapezia4.1.6Area of Compound Shapes4.1.7Area of Compound Triangle Shapes
4.2

Angles

4.2.1Angles on a Line & Around a Point4.2.2Combining Angle Facts in Triangles4.2.3Angles on Parallel Lines4.2.4Interior & Exterior Angles in Polygons
4.3

Trigonometry

4.3.1Sides in Right-Angled Triangles4.3.2Angles in Right-Angled Triangles4.3.3Exact Values of Trigonometric Ratios4.3.4Trigonometry in 3D Shapes4.3.5Using the Sine Rule4.3.6Using the Cosine Rule4.3.7Area of a Triangle Using Sine
4.4

Similarity & Congruence

4.4.1Length Scale Factors in Similar Shapes
4.5

Circle Geometry

4.5.1Calculating Circumference4.5.2Calculating Area of a Circle
4.6

Circle Theorems

4.6.1Standard Circle Theorems4.6.2Cyclic Quadrilaterals & Segment Theorems4.6.3Chord & Tangent Properties
4.7

Pythagoras

4.7.1Pythagoras' Theorem in 2D4.7.2Pythagoras’ Theorem in 3D
4.8

Surface Area

4.8.1Surface Area of Prisms4.8.2Surface Area of Spheres & Cones
4.9

Volume

4.9.1Volume of Prisms4.9.2Volume of Cylinders
4.10

Vectors

4.10.1Vectors & Column Notation4.10.2Operations with Column Vectors
4.11

Introduction to Geometry

4.11.1Geometry Basics4.11.2Geometry Basics 24.11.3Regular Polygons4.11.4Triangles & Angles4.11.5Midpoint4.11.6Bisectors & Perpendiculars4.11.7Locus4.11.8Angle rules4.11.9Parallel Angle Rules4.11.10Angles in a Polygon4.11.11Angles in a Triangle4.11.12Exam-Style Questions - Angles in Shapes4.11.13End of Topic Test - Geometry
4.12

Triangles & Quadrilaterals

4.12.1Types of Triangle4.12.2Angles & Triangles4.12.3Congruence4.12.4Similarity4.12.5Quadrilaterals
4.13

Transformations

4.13.1Types of Transformations4.13.2Invariant Points4.13.3Effects of Transformations4.13.4Exam-Style Questions - Reflection
4.14

Circle Basics

4.14.1Properties of a Circle 14.14.2Chords4.14.3Properties of a Circle 24.14.4Area of a Circle4.14.5Circumference of a Circle4.14.6Fractions of Circles4.14.7Exam-Style Questions - Circle Formulae4.14.8End of Topic Test - Triangles & Circles
4.15

Circle Theorems

4.15.1Perpendicular Bisector of a Chord4.15.2Angle at the Centre4.15.3Angle of a Semicircle4.15.4Opposite Angles in Cyclic Quadrilaterals4.15.5Tangent Lengths4.15.6Angle in the Same Segment4.15.7Alternate Segment Theorem4.15.8Exam-Style Questions - Circle Theorems4.15.9Diagnostic Misconceptions - Circumference & Radius
4.16

Measurements & Units

4.16.1Length4.16.2Area4.16.3Volume4.16.4Capacity4.16.5Time4.16.6Money4.16.7Mass4.16.8Bearings4.16.9Scale Factor & Enlargement4.16.10Exam-Style Questions - Enlargements4.16.11Diagnostic Misconceptions - Hours
4.17

Calculating Area

4.17.1Area of 2D Shapes4.17.2Exam-Style Questions - Area4.17.3End of Topic Test - Circle Theorems, Units & Area
4.18

Triangle Formulae

4.18.1Pythagoras' Theorem4.18.2Pythagoras in 3D4.18.3Trigonometry4.18.4Exact Values4.18.5Sine Rule4.18.6Cosine Rule4.18.7Area of a Triangle4.18.8Exam-Style Questions - Trigonometry & Area4.18.9Diagnostic Misconceptions - Bearings from Angles4.18.10Diagnostic Misconceptions - Pythagoras' Theorem
4.19

3D Shapes

4.19.1Types of 3D Shape4.19.2Plans & Elevations4.19.3Nets4.19.4Volume of 3D Shapes4.19.5Volume of a Cuboid4.19.6Volume of a Cylinder4.19.7Volume of a Sphere4.19.8Volume of a Hemisphere4.19.9Volume of a Cone4.19.10Volume of a Pyramid4.19.11Surface Area & Composite Shapes4.19.12Surface Area of a Cuboid4.19.13Surface Area of a Cylinder4.19.14Surface Area of a Sphere/Hemisphere4.19.15Surface Area of a Cone4.19.16Exam-Style Questions - Volume
4.20

Vectors

4.20.1Vectors Introduction4.20.2Vectors4.20.3Translation4.20.4Scalar4.20.5Exam-Style Questions - Vectors4.20.6End of Topic Test - Trigonometry, Shapes & Vectors4.20.7Diagnostic Misconceptions - Column Vectors
5

Probability

5.1

Introduction to Probability

5.1.1Randomness & Fairness5.1.2Sum of Probabilities5.1.3Relative & Expected Frequency5.1.4Frequency Tables5.1.5Using Frequency Trees5.1.6Complete Frequency Trees
5.2

Representing Probabilities

5.2.1Probability5.2.2Probability Scale5.2.3Sample Space Diagrams5.2.4Sets5.2.5Venn Diagrams5.2.6Probabilities in Venn Diagrams5.2.7Exam-Style Questions - Venn Diagrams
5.3

Probability Calculations

5.3.1Theoretical Probabilities5.3.2Multiple Events5.3.3Probability Calculations5.3.4Dependent Events5.3.5Independent Events5.3.6Probability Tree Diagram5.3.7Complex Tree Diagrams5.3.8Conditional Probability5.3.9Conditional Probability From a Two Way Table5.3.10Conditional Probability Venn Diagrams5.3.11Conditional Probability Tree Diagrams5.3.12Exam-Style Questions - Probability5.3.13End of Topic Test - Probability5.3.14Diagnostic Misconceptions - Denominator
6

Statistics

6.1

Introduction to Statistics

6.1.1Statistics Basics6.1.2Statistics Key Terms6.1.3Frequency Tables6.1.4Exam-Style Questions - Statistics & Probability6.1.5Diagnostic Misconceptions - Median6.1.6Diagnostic Misconceptions - Mode6.1.7Diagnostic Misconceptions - Mean
6.2

Charts & Diagrams

6.2.1Bar Charts6.2.2Pie Charts6.2.3Stem & Leaf Diagrams6.2.4Back to Back Stem & Leaf Diagrams6.2.5End of Topic Test - Statistics6.2.6Pictograms6.2.7Time Series Graphs6.2.8Histograms6.2.9Histograms 26.2.10Frequency Polygons6.2.11Cumulative Frequency6.2.12Cumulative Frequency 26.2.13Scatter Graphs6.2.14Scatter Graphs - Predictions6.2.15Correlation & Causation6.2.16Exam-Style Questions - Cumulative Frequency
6.3

Comparing Datasets

6.3.1Comparing Datasets6.3.2End of Topic - Datasets

Practice questions on Angle of a Semicircle

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  1. 1
    The red line is a diameter of the circle. What is angle x?Multiple choice
  2. 2
    The red line is a diameter of the circle. What is angle x?Multiple choice
  3. 3
    The red line is a diameter of the circle. What is angle x?Multiple choice
  4. 4
    The red line is a diameter of the circle. What is angle x?Multiple choice
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Angle at the Centre

Opposite Angles in Cyclic Quadrilaterals