Theorems with Chords & Tangents (Edexcel IGCSE Maths)

Circles & Chords

Spec point — 4.6B

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

• 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 OPQ = 40º.

Find the length PQ.

Answer:

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

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.)

Tangent & Radius

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 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°)

Exam Hint

• 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.

(i) Explain why angle OPB is 90°.

(ii) Find the value of $x.$

Answer:

(i)**

Final answer: 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 *

Using the fact that the internal angles in a triangle sum to 180, we can find angle $x$

$\begin{array}{rcl} x + 37 + 37 & = 180 \\ x + 74 & = 180 \end{array}$

Final answer: $x=106$

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

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 *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

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

$θ=360-205$

$θ=155^{\circ}$