Geometry & MeasuresDeep Dive

Points Dividing a Line in a Given Ratio

Part of Vectors (Geometry Proofs) · GCSE GCSE Mathematics revision

This deep dive covers Points Dividing a Line in a Given Ratio within Vectors (Geometry Proofs) for GCSE Mathematics. Revise Vectors (Geometry Proofs) in Geometry & Measures for GCSE Mathematics with 14 exam-style questions and 12 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 9 of 12 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.

Topic position

Section 9 of 12

Practice

14 questions

Recall

12 flashcards

Points Dividing a Line in a Given Ratio

If point P divides line segment AB in the ratio m:n, then:

OP = OA + [m ÷ (m + n)] × AB

This means P is m/(m+n) of the way from A to B.

Worked Example: OA = a, OB = b. Point P divides AB in ratio 2:1. Find OP.

Step 1: AB = OB - OA = b - a

Step 2: P is 2/3 of the way from A to B: AP = ⅔AB = ⅔(b - a)

Step 3: OP = OA + AP = a + ⅔(b - a) = a + ⅔b - ⅔a = ⅓a + ⅔b

Keep building this topic

Read this section alongside the surrounding pages in Vectors (Geometry Proofs). That gives you the full topic sequence instead of a single isolated revision point.

Proving Three Points Are Collinear Key Results to Remember Vectors in Geometry: Proving What You Can See Knowledge Organiser: Vectors — Geometry Proofs

Practice Questions for Vectors (Geometry Proofs)

Vector AB goes from point A to point B. Which of the following describes vector BA?

A. The same vector as AB
B. Twice the length of AB in the same direction
C. The same magnitude as AB but in the opposite direction
D. Half the length of AB in the opposite direction

1 markfoundation

A student says: 'I have shown that vector AB is parallel to vector CD, so A, B, C, D all lie on the same straight line.' Explain why the student's reasoning is incorrect.

2 marksstandard

Quick Recall Flashcards

If AB = a, what is BA?

BA = -a. Reversing the direction of travel negates the vector. In diagrams: if you travel against an arrow, write a negative sign in front of that vector.

When are two vectors parallel?

Two vectors p and q are parallel when one is a scalar multiple of the other: p = kq for some non-zero scalar k. They point in the same or exactly opposite direction.

14 questions on Vectors (Geometry Proofs) — practise free

Instant marking, adaptive difficulty, and 12 spaced repetition flashcards. Free until your GCSEs.

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Proving Three Points Are Collinear

Key Results to Remember