N.B.Nota Bene

Home  /  Standard Maths  /  MYP 5

Translations

A translation slides a shape without turning it or flipping it. All the work is in reading one small stack of two numbers: the column vector.

MYP 5Standard MathsGeometric TransformationsCriteria A · C · D~10 min read

Written by Fayek · MYP student

Sliding, not turning

A translation is the simplest transformation. Every point of the shape moves the same distance in the same direction, like sliding a book across a desk. Nothing rotates, nothing flips, and the shape keeps its exact size and orientation. The image is congruent to the original.

Translation
A transformation that moves every point of a shape by the same amount in the same direction. Size, shape and the way it faces all stay the same.

The column vector

We record a translation as a column vector: two numbers written in a stack. The top number is the movement across (right is positive, left is negative). The bottom number is the movement up or down (up is positive, down is negative).

\[ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \text{across} \\ \text{up or down} \end{pmatrix} \]

To translate a point, add the top number to its \(x\)-coordinate and the bottom number to its \(y\)-coordinate.

Top is across, bottom is up or down

A very common slip is to add the bottom number to \(x\). The top of the stack always changes the \(x\)-coordinate, the bottom always changes the \(y\)-coordinate. Keep them in that order.

A worked example

Worked example

Triangle \(ABC\) has vertices \(A(1,2)\), \(B(3,2)\) and \(C(1,5)\). Translate it by the vector \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\). Give the coordinates of the image.

1
The vector says: 4 to the right, 3 down. So add 4 to each \(x\) and subtract 3 from each \(y\).
2
\(A(1,2) \to A'(1+4,\ 2-3) = A'(5,-1)\).
3
\(B(3,2) \to B'(7,-1)\) and \(C(1,5) \to C'(5,2)\).
\(A'(5,-1)\), \(B'(7,-1)\), \(C'(5,2)\)

Describing a translation

If a question asks you to fully describe a transformation and the shape has simply slid, you must say two things: that it is a translation, and give the column vector. To find the vector, pick one point on the original and its matching point on the image, then work out how far across and how far up or down it moved.

Where this is assessed

Adding the vector to the coordinates is Criterion A (Knowing and understanding). Naming the transformation and stating the vector in the correct form is Criterion C (Communicating), and a description that leaves out the vector loses marks. Using translations on maps, grids or design problems reaches Criterion D (Applying maths in context).

Check yourself

1. Translate the point (2,7) by the vector with top -5 and bottom 1. Where does it land? +

Add the components: \(x = 2 + (-5) = -3\) and \(y = 7 + 1 = 8\). The image is \(\mathbf{(-3,8)}\).

2. A shape moves so that its point at (0,0) ends up at (6,4). What column vector describes the translation? +

It moved 6 across and 4 up, so the vector is \(\begin{pmatrix} 6 \\ 4 \end{pmatrix}\). In words, 6 right and 4 up.

3. In plain words, what movement does the vector with top 3 and bottom -2 describe? +

The top is across and the bottom is up or down, so it means 3 to the right and 2 down.

Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.