Intersecting Chord Theorem (Edexcel IGCSE Maths)

Spec point — 4.6A

Intersecting chord theorem

What is the intersecting chord theorem?

• For two chords, AB and CD that meet at point P

  • AP : PD ≡ CP : PB
  • Ratio of longer lengths (of chords) ≡ Ratio of shorter lengths (of chords)
  • This can also be written as AP × PB = CP × PD

• You do not need to know the proof of this theorem

• This theorem is closely related to similar shapes

How do I use the intersecting chord theorem to solve problems?

• If two chords intersect, you can find a missing length using the intersecting chord theorem

  • You can usually choose to solve the problem either using multiplication (AP × PB = CP × PD) or using ratio (AP : PD ≡ CP : PB)

• Carefully keep track of which distance is associated with each part of each chord

What if the lengths are algebraic?

• If any of the lengths are algebraic expressions, use the fact that AP × PB = CP × PD to form an equation and then solve it

Worked Example The diagram below shows a circle with centre O and two chords, PQ and RS.

The chords PQ and RS meet at the point T.

(a) Find the possible value(s) of $x$.

Answer:

Using the intersecting chord theorem which states that

$PT\times TQ=RT\times TS$

Fill in the information from the diagram

$( x + 2 ) 2 x = ( 3 x + 1 ) x$

Simplify

$\begin{gathered} 2 x^{2} + 4 x & = 3 x^{2} + x \\ 0 = x^{2} - 3 x \end{gathered}$

Solve by factorising

$0 = x ( x - 3 ) x = 0 \text{or} x = 3$

Consider if both answers are possible

$x$ refers to a length, so cannot be zero

$x=3$

(b) Hence, or otherwise, write down the simplified ratio of PT:RT.

Answer:

Method 1

Substitute the found value of $x$* into the expressions on the diagram for PT and RT*

$PT=x+2=5\,RT=3x+1=10$

Write as a ratio PT:RT

$5:10$

Simplify

$1:2$

Method 2

Using the ratio form of the intersecting chord theorem, and the expressions on the diagram

$\begin{gathered} P T : R T & = T S : T Q \\ ( x + 2 ) : ( 3 x + 1 ) & = ( x ) : ( 2 x ) \end{gathered}$

This shows that the ratio of PT:RT is the same as the ratio of TS:TQ

$PT:RT\,x:2x\,1:2$

$1:2$

Intersecting Chord Theorem (External)

Spec point — 4.6A

What is the external case of the intersecting chord theorem?

• The intersecting secant theorem is the mathematical name given to the external case of the intersecting chord theorem

  • A secant is a line which extends through a circle cutting the circumference at two points

• It occurs when two chords intersect outside of the circle

  • For two chords, AB and CD that extend and meet at point P outside of the circle

    • AP : PD ≡ CP : PB where AP = AB + BP and CP = CD + DP
    • Therefore (AB + BP) : PD ≡ (CD + DP) : PB

  • A more practical way to deal with most problems involving the intersecting secant theorem is

    • BP(AB + BP) = DP(CD + DP)

How do I use the intersecting secant theorem to solve problems?

• If two chords intersect outside of a circle, you can find a missing length using the intersecting secant theorem

  • Substitute the values into the multiplication formula
  • BP(AB + BP) = DP(CD + DP)

• If any of the lengths are algebraic expressions, an equation will be formed which can then be solved

Worked Example In the diagram below, A, B, C and D are points on a circle.

ABE and CDE are straight lines. AB = $x$ cm BE = 12 cm CD = 4 cm DE = 14 cm

Work out the value of $x$.

Answer:

Using the properties of Intersecting Chords (external intersection)

$AB\times AE=CD\times CE$

Or equivalently

$B E \times ( A B + B E ) = D E \times ( C D + D E )$

Substitute the values from the diagram

$12 \times ( 12 + x ) = 14 \times ( 4 + 14 )$

Solve for $x$

$\begin{gathered} 12 ( 12 + x ) & = 252 \\ 12 + x & = 21 \\ x = 9 \end{gathered}$

$x=9$

What if one of the lines is a tangent?

• A special case of the intersecting secant theorem is when one of the lines is a tangent, rather than a secant

  • A tangent touches the circumference of the circle once, rather than intersecting it

  • In this case,one of the lengths becomes zero

    • BP(AB + BP) = DP(0 + DP)
    • BP(AB + BP) = DP²

Worked Example In the diagram below, A, B, and C are points on a circle.

ABX is a straight line. YCX is a tangent to the circle. AB = 12.5 cm BE = 10 cm

Work out the length of CX.

Answer:

This is a special case of the properties of Intersecting Chords (external intersection), where one of the lines is a tangent, and therefore one of the line segments has length zero

$B X \times ( A B + B X ) = C X \times ( C X + 0 )$

Substitute the values from the diagram

$10 \times ( 12.5 + 10 ) = C X \times ( C X + 0 )$

Simplify and solve for $CX$

$\begin{gathered} 10 ( 22.5 ) & = ( C X )^{2} \\ 225 & = ( C X )^{2} \\ \sqrt{225} & = C X \\ 15 = C X \end{gathered}$

Final answer: CX = 15 cm