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IGCSE Maths · Edexcel 4XMA1 · Unit 2 Higher · Circle properties

Intersecting Chords and Secants

Edexcel’s circle properties include two length results that most textbooks skip: the intersecting chord theorems. Each comes down to one multiplication rule, and they often appear alongside the angle theorems you already know.

1Chords that cross inside the circle

If two chords AB and CD meet at a point P inside the circle, then

Intersecting chords (internal)AP × PB = CP × PD
ABCDPAPPBCPPD
Multiply the two parts of each chord
Worked example
Chords AB and CD meet at P. AP = 4 cm, PB = 6 cm and CP = 3 cm. Find PD.
  1. AP × PB = CP × PD → 4 × 6 = 3 × PD.
  2. 24 = 3 × PD → PD = 8.
PD = 8 cm

2Lines that meet outside the circle

If two lines from an external point P cut the circle at A and B, and at C and D, then

Intersecting chords (external)PA × PB = PC × PD

Each length is measured from P: PB is the whole length from P to the far point, not just AB.

ABCDPPAABPCCD
Worked example
PA = 4 cm, AB = 6 cm and PC = 5 cm. Find CD.
  1. PB = PA + AB = 4 + 6 = 10.
  2. PA × PB = PC × PD → 4 × 10 = 5 × PD → PD = 8.
  3. CD = PD − PC = 8 − 5 = 3.
CD = 3 cm
Classic error
  • Using AB instead of PB. The rule multiplies the near distance by the whole distance from P.

3A tangent and a secant

If one of the lines from P only touches the circle (a tangent at T), the two lengths PC and PD become the same, PT. So

Tangent–secantPT² = PA × PB
ABTPPAABPT
Worked example
PA = 3 cm and AB = 9 cm. Find the length of the tangent PT.
  1. PB = 3 + 9 = 12.
  2. PT² = 3 × 12 = 36.
  3. PT = 6.
PT = 6 cm

4Why it works: similar triangles

For two chords crossing at P, join AC and BD. The angles at P are vertically opposite, so they are equal. The angles at A and D stand on the same arc, so they are also equal (angles in the same segment). So triangles APC and DPB are similar, which gives

APDP = CPBP  ⇒  AP × PB = CP × PD

Exam tipEdexcel does not require formal proofs of the circle theorems, but this argument is a good similar-triangles exam question in its own right.

5Using algebra and combining with other results

Worked example
Chords AB and CD meet at P. AP = x, PB = x + 5, CP = 4 and PD = 6. Find x.
  1. x(x + 5) = 4 × 6 → x² + 5x − 24 = 0.
  2. (x + 8)(x − 3) = 0 → x = 3 or x = −8.
  3. A length is positive.
x = 3
Worked example
A diameter CD is perpendicular to a chord AB and meets it at P. AB = 16 cm and PC = 4 cm. Find the radius.
  1. A diameter perpendicular to a chord bisects it, so AP = PB = 8.
  2. AP × PB = CP × PD → 64 = 4 × PD → PD = 16.
  3. Diameter = CP + PD = 20, so radius = 10.
Radius = 10 cm

6Recap: the angle theorems

Questions in this topic often mix lengths and angles. Learn the reason phrases exactly:

TheoremReason to write
Angle at centreThe angle at the centre is twice the angle at the circumference
SemicircleThe angle in a semicircle is a right angle
Same segmentAngles in the same segment are equal
Cyclic quadrilateralOpposite angles of a cyclic quadrilateral add up to 180°
Alternate segmentThe angle between a tangent and a chord equals the angle in the alternate segment
Tangent–radiusA tangent is perpendicular to the radius at the point of contact

For full notes on the angle theorems, see the Circle Theorems page.

Common mistakes
  • For external lines, multiplying PA × AB instead of PA × PB.
  • Mixing up which parts of the chords go together. Each product uses the two parts of one chord.
  • Forgetting to square the tangent: PT², not PT.
  • Keeping the negative solution when the unknown is a length.

★Key Vocabulary Recap

Chord
A straight line joining two points on a circle.
Secant
A line that cuts a circle at two points.
Tangent
A line that touches a circle at exactly one point.
Cyclic quadrilateral
A quadrilateral with all four vertices on a circle.
Specification reference (Pearson Edexcel 4XMA1, Unit 2 Higher)
  • Use the internal and external intersecting chord properties.
  • Recognise and use angle properties of circles, including cyclic quadrilaterals and the alternate segment theorem.
  • Provide reasons, using standard geometrical statements.
Next stepReady to practise? Try the 12 exam-style questions, then check your working against the worked solutions.
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