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)
• A more practical way to deal with most problems involving the intersecting chord theorem is
  • AP × PB = CP × PD
• You do not need to know the proof of this theorem
  
  • You may be able to see a connection 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 chose to solve the problem either using multiplication (AP × PB = CP × PD) or using ratio (AP : PD ≡ CP : PB)
• Keep track carefully of which distance is associated with each part of each chord

What kind of questions involve the intersecting chord theorem?

• It is quite common to find this theorem interlinked with forming equations questions
  • Use the fact that AP × PB = CP × PD to form and equation and then solve it
  • The algebra can be made harder by having more awkward expressions for the distances involved

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 the name given to 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 carefully BP(AB + BP) = DP(CD + DP)
• Often, a quadratic equation will be formed which will need to be solved to find the missing length

What kind of questions involve the intersecting secant theorem?

• It is quite common to find this theorem interlinked with forming equations questions
  • Use the fact that BP(AB + BP) = DP(CD + DP) to form an equation and then solve it
• A special case of the intersecting secant theorem is when one of the lines is a tangent, rather than a secant
  • This means it touches the circumference of the circle once, rather than intersecting it
  • In this case, one of the lengths of the chords becomes zero and the formula changes
    • BP(AB + BP) = DP(0 + DP) becomes BP(AB + BP) = DP²