IGCSEMaths: CoreCIERevision NotesGeometryCircle TheoremsTangent & Radius

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First teaching 2021

Last exams 2024

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Tangent & Radius (CIE IGCSE Maths: Core)

Revision Note

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Jamie W

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Jamie W

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Tangent & Radius

What is a tangent?

  • A tangent to a circle is a straight line outside of the circle that touches its circumference only once
  • Tangents are the easiest thing to spot quickly in a circle theorem question as they lie outside of the circle and stand out clearly

Circle Theorem: A radius and a tangent are perpendicular

  • Most of the time, if there is a tangent in a circle theorem question it will meet a radius at the point where it touches the circumference of a circle
    • Make sure that the line the tangent meets is definitely a radius
  • This circle theorem states that a radius and a tangent meet at 90°
    • Perpendicular just means at right angles
  • When using this theorem in an exam you must use the keywords 
    • A radius and a tangent meet at right angles

Radius tangent 90, IGCSE & GCSE Maths revision notes

Examiner Tip

  • If you spot a tangent on a circle diagram, look to see if it meets a radius and add in the right angle clearly to the diagram straight away
    • In some cases just the act of doing this can earn you a mark!

Worked example

P and Q are points on the circle, centre O.

APB is a tangent to the circle at P.

tangent-and-radius-diagram-worked-example

(i)
Explain why angle OPB is 90°.
 
(ii)
Find the value of x.

(i)
 
 
Angle OPB is 90° because the angle between a tangent and a radius is 90° (and OP is a radius, and APB is a tangent).

(ii)
As angle OPB is 90°, we can find angle OPQ
 
OPQ = 90 - 53 = 37°
 
As OP and OQ are both the radius of the circle, they have the same length. This means the triangle OPQ is isosceles, so the base angles (OPQ and OQP) are equal
 
cie-igcse-core-rn-tangent-and-radius-diagram-worked-example-2-working
 
Using the fact that the internal angles in a triangle sum to 180, we can find angle x
 
table row cell x plus 37 plus 37 end cell equals 180 row cell x plus 74 end cell equals 180 end table
 
bold italic x bold equals bold 106 bold degree

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    Jamie W

    Author: Jamie W

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.