Theorems with Chords & Tangents (Edexcel IGCSE Maths A (Modular))

Revision Note

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Circles & Chords

What is a chord?

  • A chord is any straight line is a circle that joins any two points on the circumference

    • Chords of equal length are equidistant (the same distance) from the centre

Circle Theorem: The perpendicular bisector of a chord passes through the centre

  • If a line through the centre (such as a radius or diameter) goes through the midpoint of chord

    • it will bisect (cut in half) that chord at right angles to it

A circle with a radius bisecting a chord.
  • To spot this circle theorem on a diagram

    • look for a radius and see if it intersects any chords

    • or look to see if you could draw a radius that bisects a chord

  • When explaining this theorem in an exam you can use either phrase below:

    • A radius bisects a chord at right angles

    • The perpendicular bisector of a chord passes through the centre

Examiner Tips and Tricks

  • Look out for isosceles triangles formed by a chord and two radii

    • Two angles in the triangle will be equal and there will be at least one line of symmetry

Worked Example

The diagram below shows a circle with centre, O.
Two points, P and Q, lie on its circumference.
The radius of the circle is 6 cm.
Angle OPQ = 40º.

Find the length PQ.

A circle of centre, O, has two points on its circumference, P and Q. The lines OQ and PQ are drawn on. The angle OQP = 40º.

Label the radius on the diagram 6 cm

Draw a line from O to the midpoint, M, of the line PQ
The angle formed between the OM and PQ will be a right angle

A diagram showing the same circle with a straight line from the centre to the midpoint of PQ, bisecting it at a right angle.

Use SOHCAHTOA on triangle OMQ to find the length MQ

table row cell cos space 40 end cell equals cell fraction numerator M Q over denominator 6 end fraction end cell row cell 6 space cos space 40 space end cell equals cell space M Q end cell row cell M Q end cell equals cell 4.59626... end cell end table

Double MQ to find the length PQ

4.59626... cross times 2 equals 9.19253...

Round to 3 significant figures

PQ = 9.19 cm (3 s.f.)

Circles & Tangents

What is a tangent?

  • A tangent to a circle is a straight line outside of the circle that touches its circumference at exactly one point

Circle Theorem: A radius and a tangent meet at right angles

  • If a radius and a tangent meet at a point on the circumference of a circle, the angle formed between them will be 90°

    • They are perpendicular to each other

A circle with a tangent and a radius meeting at 90º.
  • When explaining this theorem in an exam you must use the keywords:

    • A radius and a tangent meet at right angles

Circle Theorem: Tangents from an external point are equal in length

  • Two tangents from the same external point are equal in length

  • This means that a kite can be formed by two tangents meeting a circle

    • The kite below has a vertical line of symmetry

      • It is formed from two congruent triangles back-to-back

    • The kite will have two right angles where the tangents meet the radii

      • You can use Pythagoras and SOHCAHTOA on each of these triangles

A circle with centre, O, and two points on the circumference, R and S. Tangents to the circle at these two points, intersect at a point outside the circle, T. OR and OS are radii. ROST forms a kite.

Examiner Tips and Tricks

  • Look for tangents in the exam and draw on the radius at right angles to see if it helps

Worked Example

Find the value of θ  in the diagram below.

A circle with centre, O, and two points on the circumference, S and R. Tangents to the circle, at S and R, meet at a point outside the circle, T. The acute angle between these two tangents at point T is 25º. The angle SOR is labelled θ.

The lines ST and RT are both tangents to the circle
They meet the two radii on the circumference at the points and T

Angle TSO = angle TRO = 90°
A radius and a tangent meet at right angles

Use vertically opposite angles to find the value of angle RTS

Angle RTS = 25°
Vertically opposite angles

Mark these angles clearly on the diagram

The diagram of the circle with two tangents as before but with the angles OST and ORT marked as being perpendicular and the angle STR labelled 25º.

Angles in a quadrilateral add up to 360°
Use this to form an equation for θ

theta space plus space 90 space plus space 90 space plus space 25 space equals space 360
Angles in a quadrilateral sum to 360º

Simplify

theta space plus space 205 space equals space 360

Solve

theta space equals space 360 space minus space 205 space

bold space bold italic theta bold space bold equals bold space bold 155 bold degree

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

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.