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Maths Gcse Edexcel Higher

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  1. Scatter-Graphs-And-Correlation Edexcel Higher
    2 主题
  2. Cumulative-Frequency-And-Box-Plots Edexcel Higher
    4 主题
  3. Histograms Edexcel Higher
    3 主题
  4. Statistical-Diagrams Edexcel Higher
    7 主题
  5. Averages-Ranges-And-Data Edexcel Higher
    8 主题
  6. Combined-And-Conditional-Probability Edexcel Higher
    3 主题
  7. Tree-Diagrams Edexcel Higher
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  8. Simple-Probability-Diagrams Edexcel Higher
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  9. Transformations Edexcel Higher
    5 主题
  10. Vectors Edexcel Higher
    6 主题
  11. 3D-Pythagoras-And-Trigonometry Edexcel Higher
    1 主题
  12. Sine-Cosine-Rule-And-Area-Of-Triangles Edexcel Higher
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  13. Pythagoras-And-Trigonometry Edexcel Higher
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  14. Area-And-Volume-Of-Similar-Shapes Edexcel Higher
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  15. Congruence-Similarity-And-Geometrical-Proof Edexcel Higher
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  16. Volume-And-Surface-Area Edexcel Higher
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  17. Circles-Arcs-And-Sectors Edexcel Higher
    2 主题
  18. Area-And-Perimeter Edexcel Higher
    4 主题
  19. Circle-Theorems Edexcel Higher
    7 主题
  20. Bearings-Scale-Drawing-Constructions-And-Loci Edexcel Higher
    5 主题
  21. Angles-In-Polygons-And-Parallel-Lines Edexcel Higher
    3 主题
  22. Symmetry-And-Shapes Edexcel Higher
    6 主题
  23. Exchange-Rates-And-Best-Buys Edexcel Higher
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  24. Standard-And-Compound-Units Edexcel Higher
    5 主题
  25. Direct-And-Inverse-Proportion Edexcel Higher
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  26. Problem-Solving-With-Ratios Edexcel Higher
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  27. Ratios Edexcel Higher
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  28. Sequences Edexcel Higher
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  29. Transformations-Of-Graphs Edexcel Higher
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  30. Graphing-Inequalities Edexcel Higher
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  31. Solving-Inequalities Edexcel Higher
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  32. Real-Life-Graphs Edexcel Higher
    4 主题
  33. Estimating-Gradients-And-Areas-Under-Graphs Edexcel Higher
    2 主题
  34. Equation-Of-A-Circle Edexcel Higher
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  35. Graphs-Of-Functions Edexcel Higher
    6 主题
  36. Linear-Graphs Edexcel Higher
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  37. Coordinate-Geometry Edexcel Higher
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  38. Functions Edexcel Higher
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  39. Forming-And-Solving-Equations Edexcel Higher
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  40. Iteration Edexcel Higher
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  41. Simultaneous-Equations Edexcel Higher
    2 主题
  42. Quadratic-Equations Edexcel Higher
    4 主题
  43. Linear-Equations Edexcel Higher
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  44. Algebraic-Proof Edexcel Higher
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  45. Rearranging-Formulas Edexcel Higher
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  46. Algebraic-Fractions Edexcel Higher
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  47. Completing-The-Square Edexcel Higher
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  48. Factorising Edexcel Higher
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  49. Expanding-Brackets Edexcel Higher
    3 主题
  50. Algebraic-Roots-And-Indices Edexcel Higher
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  51. Introduction Edexcel Higher
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  52. Using-A-Calculator Edexcel Higher
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  53. Surds Edexcel Higher
    2 主题
  54. Rounding-Estimation-And-Bounds Edexcel Higher
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  55. Fractions-Decimals-And-Percentages Edexcel Higher
    3 主题
  56. Simple-And-Compound-Interest-Growth-And-Decay Edexcel Higher
    4 主题
  57. Percentages Edexcel Higher
    3 主题
  58. Fractions Edexcel Higher
    4 主题
  59. Powers-Roots-And-Standard-Form Edexcel Higher
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  60. Prime-Factors-Hcf-And-Lcm Edexcel Higher
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  61. Number-Operations Edexcel Higher
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Exam code:1MA1

Alternate segment theorem

Circle theorem: The alternate segment theorem

  • The angle between a chord and a tangent is equal to the angle in the alternate segment

Alternate Segment Theorem, IGCSE & GCSE Maths revision notes

  • To spot this circle theorem on a diagram

    • look for a cyclic triangle

      • where all three vertices of the triangle lie on the circumference

    • one vertex of the triangle meets a tangent

  • To identify which angles are equal

    • mark the angle between the tangent and the side of the cyclic triangle

    • the angle inside the triangle at the corner opposite the side of the triangle that forms the first angle is the equal angle

  • When explaining this theorem in an exam you can just say the phrase:

    • The Alternate segment theorem

Examiner Tips and Tricks

  • Look for cyclic triangles and tangents in busy diagrams

    • Questions involving the alternate segment theorem frequently appear in exams!

Worked Example

Find the value of x.

A circle with centre, O, has three points on the circumference, P, Q and R. A tangent meets the circumference of the circle at point R. PQR and QOR are triangles. Angle QOR = (5x-2) and the angle between QR and the tangent is (2x+5).

Identify the cyclic quadrilateral (triangle in the circle with all three vertices at the circumference)

One vertex of this triangle meets a tangent at point R
The angle between one of its sides (QR) and the tangent is given
Find the angle inside the triangle, opposite to the same side (QR)

Angle between QR and the tangent = Angle RQP = open parentheses 2 x plus 5 close parentheses
Alternate segment theorem

Same circle diagram with the angle RPQ marked as (2x+5).

Notice that angle RPQ and angle ROQ both come from the same two points on the circumference

Angle ROQ = <img alt=”2 open parentheses 2 x plus 5 close parentheses” data-mathml=”<math ><semantics><mrow><mn>2</mn><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true,”toolbar”:”<toolbar ref=’general’><tab ref=’general’><removeItem ref=’setColor’/><removeItem ref=’bold’/><removeItem ref=’italic’/><removeItem ref=’autoItalic’/><removeItem ref=’setUnicode’/><removeItem ref=’mtext’ /><removeItem ref=’rtl’/><removeItem ref=’forceLigature’/><removeItem ref=’setFontFamily’ /><removeItem ref=’setFontSize’/></tab></toolbar>”}</annotation></semantics></math>” data-type=”working” height=”22″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2222%22%20width%3D%2267%22%20wrs%3Abaseline%3D%2216%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmfenced%3E%3Cmrow%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfenced%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math117e62166fc8586dfa4d1bc0e17’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADRjdnQgDVUNBwAAAVAAAAA6Z2x5ZoPi2VsAAAGMAAAAoWhlYWQQC2qxAAACMAAAADZoaGVhCGsXSAAAAmgAAAAkaG10eE2rRkcAAAKMAAAACGxvY2EAHTwYAAAClAAAAAxtYXhwBT0FPgAAAqAAAAAgbmFtZaBxlY4AAALAAAABn3Bvc3QB9wD6AAAEYAAAACBwcmVwa1uragAABIAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACAAAAAEAAQAAQAAACv%2F%2FwAAACv%2F%2F%2F%2FWAAEAAAAAAAABVAMsAIABAABWACoCWAIeAQ4BLAIsAFoBgAKAAKAA1ACAAAAAAAAAACsAVQCAAKsA1QEAASsABwAAAAIAVQAAAwADqwADAAcAADMRIRElIREhVQKr%2FasCAP4AA6v8VVUDAAABAIAAVQLVAqsACwBJARiyDAEBFBMQsQAD9rEBBPWwCjyxAwX1sAg8sQUE9bAGPLENA%2BYAsQAAExCxAQbksQEBExCwBTyxAwTlsQsF9bAHPLEJBOUxMBMhETMRIRUhESMRIYABAFUBAP8AVf8AAasBAP8AVv8AAQAAAAAAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%2F7YEgAABAAAAAAAAAAAAAAAAAAAAAgNSAFUDVgCAAAAAAAAAACgAAAChAAEAAAACAF4ABQAAAAAAAgCABAAAAAAABAAA3gAAAAAAAAAVAQIAAAAAAAAAAQASAAAAAAAAAAAAAgAOABIAAAAAAAAAAwAwACAAAAAAAAAABAASAFAAAAAAAAAABQAWAGIAAAAAAAAABgAJAHgAAAAAAAAACAAcAIEAAQAAAAAAAQASAAAAAQAAAAAAAgAOABIAAQAAAAAAAwAwACAAAQAAAAAABAASAFAAAQAAAAAABQAWAGIAAQAAAAAABgAJAHgAAQAAAAAACAAcAIEAAwABBAkAAQASAAAAAwABBAkAAgAOABIAAwA

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