Theorems with Chords & Tangents (Edexcel IGCSE Maths)

Circles & Chords

Spec point — 4.6B

What is a chord?

• A chord is any straight line in 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 a chord

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

• 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

Exam Hint 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 OQP = 40º.

Find the length PQ.

Answer:

Label the radius on the diagram as 6 cm

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

Use SOHCAHTOA on triangle OMQ to find the length* MQ*

$\begin{gathered} \cos 40 & = \frac{M Q}{6} \\ 6 \cos 40 & = M Q \\ M Q & = 4.59626... \end{gathered}$* *

Double MQ to find the length PQ

$4.59626...\times2=9.19253...$

Round to 3 significant figures

Final answer: PQ = 9.19 cm (3 s.f.)

Circles & Tangents

Spec point — 4.6B

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

• 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

Exam Hint 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.

Answer:

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

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

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

$θ+90+90+25=360$ Angles in a quadrilateral sum to 360º

Simplify

$θ+205=360$

Solve

$θ=360-205$

$θ=155^{\circ}$