# Intersecting Chord Theorem

## 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

#### Exam Tip

• If you do not like the capital letter notation used you can rename the lengths of the chord using single letters (see the diagram above)
• The multiplication version of the theorem is easier to remember and work with but you may be asked questions about ratios too

## 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
• 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

#### Worked example

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

ABE and CDE  are straight lines.
BE  = 12 cm
CD  = 4 cm
DE  = 14 cm
Work out the length of AB.

Using the properties of Intersecting Chords (external intersection), we know that

Or equivalently,

Substitute the values given in the question.

Simplify.

Divide both sides by 12.

Solve by subtracting 9 from both sides.

9 cm

(b)
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.

This is a special case of the properties of Intersecting Chords (external intersection).

Substitute the numbers given in the question.

Simplify.

Solve for CX by taking the positive square root (CX is a length and cannot be negative).

15 cm

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