Sheet № 159 · Foundation + Higher · AQA · Edexcel · OCR
Translations and Column Vectors –
Translations are one of the four transformations you need to master for GCSE Maths. A translation slides a shape from one position to another without rotating, reflecting, or resizing it. The movement is described using a column vector. This guide covers how to read and write column vectors, apply translations, and describe them fully in
§Key definitions
Question:
Translate the triangle with vertices A(2, 3), B(5, 3), and C(5, 6) by the vector (3 over −2).
Answer:
The translated vertices are A'(5, 1), B'(8, 1), C'(8, 4).
Q1 (Foundation):
Translate the point (4, 2) by the vector (−3 over 5).
Q2 (Foundation):
A shape is translated so that the point (6, 1) maps to (2, 4). What is the translation vector?
Q3 (Higher):
A shape is translated by the vector (a over −1). A vertex at (3, 5) maps to (−2, 4). Find the value of a.
§Formulas to memorise
Column vector (a over b) means move a units horizontally and b units vertically
New coordinates: (x + a, y + b) where the vector is (a over b)
Read the column vector. — The top number is the horizontal shift; the bottom number is the vertical shift.
Apply the vector to each vertex. — Add the top number to each x-coordinate and the bottom number to each y-coordinate.
Plot the new vertices — and draw the translated shape.
To describe a translation — , find the column vector by subtracting the original coordinates from the image coordinates.
Worked example
Translate the triangle with vertices A(2, 3), B(5, 3), and C(5, 6) by the vector (3 over −2).
Working:
⚠ Common mistakes
- ✗Mixing up the horizontal and vertical components. The top number in the column vector is always horizontal (left/right) and the bottom number is always vertical (up/down). Swapping them moves the shape in the wrong direction.
- ✗Getting the sign wrong. Positive top means right; negative top means left. Positive bottom means up; negative bottom means down. A sign error flips the direction of the translation.
- ✗Not describing the transformation correctly. You must say "Translation by the vector ..." and give the column vector. Writing "the shape moves 3 right and 2 down" without a vector may lose marks.
✦ Exam tips
- →Always describe a translation using a column vector — this is the expected format in GCSE exams.
- →Check your answer by verifying that all vertices have been shifted by the same vector.
- →If you need to find the vector from a diagram, pick one vertex and count the squares right/left and up/down to its image.
- →Translations preserve congruence — the image is identical to the object.