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Grade 8 · Higher · free GCSE maths worksheet

Intersecting chords

Edexcel IGCSE

AX times XB = CX times XD for chords crossing inside or outside a circle.

10 original questions, 29 marks. Type your answers below for an instant mark and a score, or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

How to do it

  1. Name the two chords and the point where they cross, then list the four lengths
  2. Crossing inside the circle at X: AX × XB = CX × XD
  3. Meeting outside the circle at P: PA × PB = PC × PD, every length measured from P
  4. If PB is not given, add it up first: PB = PA + AB
  5. Put the numbers in, then divide (or solve the quadratic) to find the unknown

Watch out: Outside the circle every length starts at P: PB is the whole line PA + AB, not just the chord AB.

Worked example

Chords AB and CD of a circle, when extended, meet at the point P outside the circle. The points P, A, B lie in that order on one straight line and the points P, C, D lie in that order on the other.

PA = 3 cm, AB = 5 cm and PC = 4 cm.

Work out the length of CD.

Lengths from P: PB is the whole line

PB=3 + 5 = 8

External chords: PA × PB = PC × PD

3 × 8=4 × PD
24=4 × PD
÷4PD=6÷4
CD=PD - PC = 6 - 4

CD = 2 cm

The questions

1

Chords AB and CD of a circle cross at the point X inside the circle.

AX = 3 cm, XB = 8 cm and CX = 4 cm.

Work out the length of XD.

2 marks
Show answer
6
2

Chords AB and CD of a circle cross at the point X inside the circle.

AX = 5 cm, XB = 6 cm and XD = 4 cm.

Work out the length of CX.

2 marks
Show answer
7.5
3

Chords AB and CD of a circle, when extended, meet at the point P outside the circle. The points P, A, B lie in that order on one straight line and the points P, C, D lie in that order on the other.

PA = 4 cm, PB = 10 cm and PC = 5 cm.

Work out the length of PD.

2 marks
Show answer
8
4

Chords AB and CD of a circle, when extended, meet at the point P outside the circle. The points P, A, B lie in that order on one straight line and the points P, C, D lie in that order on the other.

PA = 3 cm, PB = 14 cm and PD = 12 cm.

Work out the length of PC.

2 marks
Show answer
3.5
5

Chords AB and CD of a circle, when extended, meet at the point P outside the circle. The points P, A, B lie in that order on one straight line and the points P, C, D lie in that order on the other.

PA = 5 cm, AB = 7 cm and PC = 4 cm.

Work out the length of the chord CD.

3 marks
Show answer
11
6

A circular pond has four gates, A, B, C and D, on its edge.

A straight path from gate A to gate B and a straight path from gate C to gate D cross each other at the point X on the pond.

AX = 9 m, XB = 10 m and CX = 6 m.

Work out the total length of the path from C to D. Give your answer in metres.

3 marks
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21
7

AB is a diameter of a circle. The chord CD crosses AB at the point X.

AX = 3 cm, CX = 4 cm and XD = 9 cm.

  1. Work out the length of XB.
  2. Work out the radius of the circle.
3 marks
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a) 12 b) 7.5
8

Chords AB and CD of a circle cross at the point X inside the circle.

AX = x cm, XB = (x + 3) cm, CX = 2 cm and XD = 9 cm.

Find the value of x.

4 marks
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x = 3
9

The straight lines AB and CD meet at the point P. The points P, A, B lie in that order on one of the lines and the points P, C, D lie in that order on the other.

PA = 5 cm, AB = 7 cm, PC = 6 cm and PD = 10 cm.

Lina says that the four points A, B, C and D all lie on the same circle.

Is Lina correct? You must show your working.

3 marks
Show answer
Yes. PB = 5 + 7 = 12, so PA × PB = 5 × 12 = 60 and PC × PD = 6 × 10 = 60. The products are equal, which is exactly the intersecting chords property for chords meeting outside a circle, so A, B, C and D lie on one circle.
10

Chords AB and CD of a circle, when extended, meet at the point P outside the circle. The points P, A, B lie in that order on one straight line and the points P, C, D lie in that order on the other.

PA = 3 cm, AB = 9 cm, PC = x cm and CD = (x + 1) cm.

Work out the length of PD.

5 marks
Show answer
9

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Every question here is original and written by Robinson Tuition. This topic is on the Pearson Edexcel International GCSE (4MA1) and is not examined on the UK GCSE; the grade is our estimate against the Edexcel grade descriptors. This is not an exam-board paper.

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