Intersecting Chord Theorem (Edexcel IGCSE Maths A (Modular))

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

Intersecting Chord Theorem (External)

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

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