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The Alternate Segment Theorem (Cambridge (CIE) IGCSE International Maths)

Revision Note

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 2Edexcel Maths A: HigherCambridge (CIE) Maths: Extended

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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 = 2 open parentheses 2 x plus 5 close parentheses
Angle at the centre is twice the angle at the circumference

Form an equation using the two expressions for angle ROQ

2 left parenthesis 2 x plus 5 right parenthesis equals 5 x minus 2

Expand the brackets and solve

table row cell 4 x plus 10 end cell equals cell 5 x minus 2 end cell row 12 equals x end table

bold italic x bold equals bold 12

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.