Intersecting chords and secants

International GCSE Higher uses two facts about chords that cross, inside or outside a circle. Both say that two products are equal.

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Chords crossing inside

Chords AB and CD cross at X inside the circle.
AX × XB = CX × XD
AX = 4, XB = 6, CX = 3: 4 × 6 = 3 × XD, so XD = 8.

Chords meeting outside

Lines through P cut the circle at A and B, and at C and D (measure from P).
PA × PB = PC × PD
PA = 4, AB = 5 (so PB = 9), PC = 3: 4 × 9 = 3 × PD, so PD = 12 and CD = 9.

With a tangent

If the line from P is a tangent touching at T, the two points become one:
PT2 = PA × PB
PA = 4, PB = 9: PT2 = 36, so PT = 6.
Always measure every length from P, not from A to B.
Worked example

Chords AB and CD cross at X. AX = 4 cm, XB = 6 cm and CX = 3 cm. Find XD.

  1. AX × XB = CX × XD
  2. 4 × 6 = 3 × XD
  3. XD = 8 cm

Answer: 8 cm

Key idea

Inside: AX × XB = CX × XD. Outside: PA × PB = PC × PD, measured from P. Tangent: PT2 = PA × PB.

Quick check

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Extra practice

2 more questions

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  1. Chords AB and CD cross at X. AX = 5 cm, XB = 8 cm and CX = 2 cm. Find XD.

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    Answer: 20 cm

    5 × 8 = 2 × XD, so XD = 20 cm.

  2. From P, one line cuts a circle at A and B, another at C and D. PA = 6 cm, AB = 8 cm and PC = 3 cm. Find PD.

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    Answer: 28 cm

    PB = 14, so 6 × 14 = 3 × PD and PD = 28 cm.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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