Section Formula: Dividing a Line Segment in a Given Ratio

What is the section formula?

If point A has coordinates (x₁, y₁) and point B has coordinates (x₂, y₂), the point P that divides AB in the ratio m : n has coordinates:

P = ( (mx₂ + nx₁) / (m + n) , (my₂ + ny₁) / (m + n) )

This is called internal division — P lies between A and B on the segment. The formula is a weighted average: P is closer to B when m is larger, and closer to A when n is larger.

How do you use the section formula step by step?

1. Label the endpoints. Write A = (x₁, y₁) and B = (x₂, y₂) clearly. Identify m : n from the question — m is the ratio part from A to P, and n is the ratio part from P to B.

2. Apply the formula for the x-coordinate. Compute (mx₂ + nx₁) ÷ (m + n).

3. Apply the formula for the y-coordinate. Compute (my₂ + ny₁) ÷ (m + n).

4. Write the answer as coordinates. Combine the two results into (x, y).

5. Check with a sketch. Draw A, B and your answer P. Measure (or estimate) whether P looks roughly correct for the stated ratio.

Worked example

A = (2, 1) and B = (8, 7). Find the point P that divides AB in the ratio 1 : 2.

Here m = 1, n = 2 (P is one-third of the way from A to B).

x-coordinate of P: (1 × 8 + 2 × 2) / (1 + 2) = (8 + 4) / 3 = 12/3 = 4

y-coordinate of P: (1 × 7 + 2 × 1) / (1 + 2) = (7 + 2) / 3 = 9/3 = 3

So P = (4, 3).

Check: the distance from A(2,1) to P(4,3) is √(4+4) = √8. The distance from P(4,3) to B(8,7) is √(16+16) = √32 = 2√8. Ratio AP : PB = √8 : 2√8 = 1 : 2 ✓

What if the ratio is 2 : 3?

How does the section formula relate to vectors?

In vector notation, if a and b are the position vectors of A and B, then the position vector of P dividing AB in ratio m : n is:

p = (na + mb) / (m + n)

This is the same formula expressed using vectors. GCSE questions may present this as a vector proof or ask you to state the position vector rather than coordinates.

What is the midpoint formula?

The midpoint formula is a special case of the section formula with m = n = 1 (ratio 1:1). Substituting: midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). If you recognise that dividing in ratio 1:1 gives the midpoint, you can check your section-formula answer: a ratio of 1:3 should give a point one-quarter of the way from A to B, which you can verify against the midpoint.

Frequently asked questions

Does the order of the ratio matter?

Yes. Ratio m : n means the segment from A to P has length in proportion m, and from P to B in proportion n. Swapping to n : m gives a different point P that is the mirror image of the correct answer about the midpoint. Always re-read the question to confirm which end the ratio is measured from.

What if the coordinates are negative?

Substitute the negative values directly into the formula and be careful with signs. For example, if x₁ = −3 and m = 2, the term mx₁ = 2 × (−3) = −6. Using brackets around negative values avoids sign errors.

Is external division on the GCSE syllabus?

External division (where P lies outside segment AB) is not typically required at GCSE. It appears at A-level. At GCSE you only need internal division, where m and n are both positive and P lies between A and B.

How many marks does this topic carry?

Section formula / dividing-in-a-ratio questions typically appear in the 3–5 mark range on Higher papers. They are usually part of a multi-part vector or coordinate geometry question, so accuracy on this step carries marks into subsequent parts.

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