Divide a directed segment in the stated internal ratio
Read AP:PB in the correct order · Use the fraction m/(m+n) from A toward B · Verify a ratio point with its two displacements
Section Paper 4 · Historical Extended · calculator
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Midpoints
- Ratios and position vectors
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Read AP:PB in the correct order
- Use the fraction m/(m+n) from A toward B
- Verify a ratio point with its two displacements
Start at the named endpoint
For internal division AP:PB=m:n with positive m,n, P=A+[m/(m+n)](B−A). The fraction is the part AP of the whole AB, not m/n. Equivalently P=(nA+mB)/(m+n): the endpoint weights are crossed because movement begins at A.
For A(0,1), B(8,5), ratio 1:3 gives fraction 1/4 and P(2,2). Its remaining displacement to B is three times its displacement from A. This is not the midpoint unless m=n.
Direction and conditions
Changing the starting endpoint while keeping the written ratio can change the point. Positive m,n place P strictly between the endpoints. Negative ratios do not describe this internal-division rule without a different stated convention; do not apply it blindly to an undefined ratio or a zero total.
Sources for this lesson
Pause and explain
Which fraction of AB is AP when AP:PB=1:3?
Worked example
For A(0,1), B(8,5), find P with AP:PB=1:3.
Show the worked solution
- B−A=(8,4), and one quarter is (2,1).
- Add this to A to obtain (2,2).
Answer P(2,2).
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A(−2,1), B(6,5). Find P with AP:PB=1:3.
Give me a hint
Move one quarter of the displacement from A.
Compare my reasoning
- B−A=(8,4); one quarter is (2,1).
- P=(−2,1)+(2,1)=(0,2).
P(0,2).
Look for these in your work
- Use the fraction of the whole
- Start at A
A(1,−2), B(7,4). Find P with AP:PB=2:1, then Q with BQ:QA=2:1.
Give me a hint
Use the named start for each two-thirds movement.
Compare my reasoning
- P=A+(2/3)(B−A)=(5,2).
- Q=B+(2/3)(A−B)=(3,0).
P(5,2); Q(3,0).
Look for these in your work
- Respect direction
- Check both ratios
A(0,0), B(12,8). A student calls P(9,6) the point with AP:PB=1:3. Find its actual ratio and the correct point for 1:3.
Give me a hint
Compare the fraction travelled from A.
Compare my reasoning
- P is three quarters along AB, so AP:PB=3:1.
- The required one-quarter point is (3,2).
P has ratio 3:1; the required 1:3 point is (3,2).
Look for these in your work
- Check the ratio claim
- Correct the starting fraction
Common mistakes
- The fraction of the whole is m/(m+n), not m/n.
- Keep the named start and ratio order.
What can you explain now?
Which fraction of AB is AP when AP:PB=1:3?
Compare with the explanation
1/4.
The complete segment has 1+3=4 ratio parts.
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Make it stick
Which fraction of AB is AP when AP:PB=1:3?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.