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MathidiaMathematics Academy
Vectors and geometric proof
Lesson 48 of 75
Standard practice

Dividing lines in a ratio

Objective. Locate internal division points using vectors.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Dividing lines in a ratio is developed here as connected GCSE mathematics, not a memorised shortcut. Represent position, movement and collinearity algebraically. The lesson moves from a precise representation and key definitions to fluent procedure, linked reasoning and an unfamiliar application. Learners compare possible methods, state restrictions and assumptions, preserve exact values where appropriate, and justify accuracy, notation and units before evaluating whether the result is mathematically and contextually reasonable.

Worked example

How can collinearity be shown with vectors?

one displacement is a sca…
Illustrated card showing one displacement is a scalar multiple of another
  1. 1Translate the information into a labelled diagram, table, graph or algebraic statement and identify every constraint.
  2. 2Select the most efficient GCSE method, explain why its conditions apply, and keep exact values until approximation is requested.
  3. 3Carry out each transformation on a separate justified line, tracking signs, units, domains and required accuracy.
  4. 4Verify with substitution, an inverse method, estimation or a second representation. The checked result is one displacement is a scalar multiple of another.

Answer: one displacement is a scalar multiple of another

Concept mastery

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Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

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Common mistake. A common error in dividing lines in a ratio is applying a familiar rule before checking its conditions, domain or direction. Name the structure first, retain exact values, and test the final result against every original constraint.

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