IGCSEMathsEdexcelMaths AFoundationRevision Notes4. Geometry & TrigonometryCircles, Arcs & SectorsTheorems with Chords & Tangents

Theorems with Chords & Tangents (Edexcel IGCSE Maths A) : Revision Note

Jamie Wood

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Edexcel Maths A: FoundationCambridge (CIE) International Maths: CoreCambridge (CIE) Maths: Core

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.)

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 Tips and Tricks

  • 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

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 S 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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Jamie Wood

Author: Jamie Wood

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.

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