# The point dividing a line in a given ratio

> Edexcel International GCSE Further Pure Mathematics · 4PM1
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s8-the-point-dividing-a-line/

This guide teaches you to find coordinates of a point dividing a line segment in a given ratio, including the midpoint special case, aligned to Edexcel IGCSE Further Pure Math 4PM1 syllabus section 8B.

**Prerequisites:** [Ability to plot and identify points on a Cartesian coordinate plane](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s8-cartesian-coordinates-basics/); Understanding of ratio notation and simplification

## Learning objectives

- Recall the section formula for a point dividing a line segment in a given ratio m:n
- Apply the formula to find unknown coordinates of division points, including the midpoint special case
- Avoid common errors with the order of ratio values and coordinates when solving exam questions

## The Section Formula for Internal Division

When a point divides the line segment joining points $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m:n$, we use the section formula to find its coordinates. This formula is not provided in your exam, so you must memorize it.

**Section Formula** — Gives coordinates of point P dividing segment AB from A to B in ratio $m:n$, where $AP:PB = m:n$

*Notation:* P\left(\frac{nx_1 + mx_2}{m+n}, \frac{ny_1 + my_2}{m+n}\right)

*Example:* For A(1, 2) and B(4, 6), ratio 1:2 gives $P\left(\frac{2*1 + 1*4}{3}, \frac{2*2 + 1*6}{3}\right) = \left(2, \frac{10}{3}\right)$

**Worked example:** Find the coordinates of the point P that divides the line segment joining A(3, 5) and B(9, 2) in the ratio 2:1, where $AP:PB = 2:1$.

1. Identify values for the formula: $x_1 = 3, y_1 = 5$ (coordinates of A), $x_2 = 9, y_2 = 2$ (coordinates of B), $m = 2, n = 1$
2. $$x_P = \frac{n x_1 + m x_2}{m + n} = \frac{1*3 + 2*9}{2+1} = \frac{3 + 18}{3} = 7$$
3. $$y_P = \frac{n y_1 + m y_2}{m + n} = \frac{1*5 + 2*2}{3} = \frac{5 + 4}{3} = 3$$
4. The coordinates of P are (7, 3)

> **tip**
>
> Always label the ratio with the endpoints first to avoid mixing m and n values: m corresponds to the segment closest to the second endpoint B, not A.

*Calculator:* allowed

## Special Case: Midpoint of a Line Segment

The midpoint of a line segment is the point that divides the segment in the ratio 1:1, so we can substitute $m=1$ and $n=1$ into the section formula to get a simplified midpoint formula.

**Midpoint Formula** — Gives the coordinates of the midpoint of the line segment joining points $(x_1, y_1)$ and $(x_2, y_2)$

*Notation:* M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

*Example:* Midpoint of (2, 4) and (6, 8) is $\left(\frac{2+6}{2}, \frac{4+8}{2}\right) = (4, 6)$

**Worked example:** Find the midpoint of the line segment joining C(-2, 7) and D(4, -3).

1. Identify values: $x_1 = -2, y_1 = 7$, $x_2 = 4, y_2 = -3$
2. $$x_M = \frac{-2 + 4}{2} = 1$$
3. $$y_M = \frac{7 + (-3)}{2} = 2$$
4. Midpoint M has coordinates (1, 2)

**Check your understanding**

1. What ratio corresponds to the midpoint of a line segment?

   - 1:2
   - 2:1
   - 1:1
   - 3:1

   *Answer:* 1:1

   *Why:* Correct! The midpoint splits the segment into two equal parts, so the ratio of the two resulting segments is 1:1.

*Calculator:* allowed

## Solving for Unknown Endpoints or Ratios

You may be given the coordinates of the division point and one endpoint, and asked to find the other endpoint, or given both endpoints and the division point to find the ratio. Rearrange the section formula to solve for these unknown values.

**Worked example:** The point P(5, 3) divides the segment joining A(2, 9) and B(x, y) in the ratio 1:2 ($AP:PB = 1:2$). Find the coordinates of B.

1. Substitute known values into the x-coordinate section formula: $m=1, n=2$, $x_1=2$, $x_P=5$
2. $$5 = \frac{2*2 + 1*x}{1+2} = \frac{4 + x}{3}$$
3. Multiply both sides by 3: $15 = 4 + x \rightarrow x = 11$
4. $$Repeat for y-coordinate: $3 = \frac{2*9 + 1*y}{3} = \frac{18 + y}{3}$$$
5. Multiply both sides by 3: $9 = 18 + y \rightarrow y = -9$
6. Coordinates of B are (11, -9)

> **Exam tip**
>
> Write out the formula explicitly for each coordinate when solving for unknowns, to avoid mixing m and n values even under exam time pressure.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Swapping m and n when applying the section formula, e.g. multiplying m by the first point's coordinates instead of the second.
  - Why it fails: The ratio $m:n$ refers to $AP:PB$, so m corresponds to the segment closer to the second endpoint B, not A, leading to inverted weights if swapped.
  - Correct: Label the ratio with the endpoints first: if $AP:PB = m:n$, n multiplies A's coordinates, m multiplies B's coordinates before dividing by $m+n$.
- **Wrong:** Forgetting to divide by $m + n$ when calculating coordinates.
  - Why it fails: The section formula is a weighted average of the two endpoints, so omitting the division by the total ratio parts gives an incorrect scaled coordinate.
  - Correct: After calculating the weighted sum of x and y coordinates, always divide by $m + n$, even for the midpoint formula (divide by $2 = 1+1$).
- **Wrong:** Using the wrong order of endpoints when the ratio is given for B to A instead of A to B.
  - Why it fails: If the ratio is $PB:AP = m:n$ instead of $AP:PB$, the values of m and n are swapped, leading to completely incorrect coordinates.
  - Correct: Rewrite the ratio explicitly in the order of the first named endpoint to the second named endpoint before substituting into the formula.
- **Wrong:** Assuming the formula works for points outside the line segment AB.
  - Why it fails: All ratio division questions in 4PM1 only test internal division, with the point lying between the two endpoints; external division is not in the syllabus.
  - Correct: Confirm the division point is between A and B before applying the standard section formula, as external division will not be tested.

## Cheatsheet

| Scenario | Formula | Key Notes |
| --- | --- | --- |
| Point dividing AB in ratio $m:n$ ($AP:PB = m:n$) | $\left(\frac{n x_1 + m x_2}{m+n}, \frac{n y_1 + m y_2}{m+n}\right)$ | A = $(x_1, y_1)$, B = $(x_2, y_2)$ |
| Midpoint of AB | $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ | Special case $m:n = 1:1$ |
| Solve for unknown endpoint/ratio | Rearrange formula to isolate unknown | Write one separate equation for x and y coordinates |

## What's next

Now that you can find points dividing lines in a given ratio, you are ready to move on to other core coordinate geometry topics in Edexcel IGCSE Further Pure Math. The next topic covers calculating the gradient of a line segment, which you will use alongside ratio division to solve full straight-line geometry problems. You can also practice applying this formula to structured exam questions to build speed and accuracy before your assessment. Make sure you memorize the section formula fully, as it is not provided in the exam formula booklet and will be required for nearly all coordinate geometry questions in section 8 of the syllabus.

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