Syllabus Edition

First teaching 2023

First exams 2025

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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

Circle Theorem: A radius and a tangent are perpendicular

  • This circle theorem states that a radius and a tangent meet at right angles (90°)
    • This may also be described as being perpendicular to each other
  • When using this theorem in an exam you must use the keywords 
    • A radius and a tangent meet at right angles (or 90°)

Radius and tangent are perpendicular circle theorem

Examiner Tip

  • If you spot a tangent on a circle diagram, look to see if it meets a radius and label the right angle on the diagram
    • In some cases just doing this can earn you a mark!
  • If you think you have spotted this circle theorem in a question, make sure it is a radius that meets the tangent, and not a chord
    • A radius passes through the centre of the circle

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 attributes columnalign right center left columnspacing 0px end attributes 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

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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.