Translation in Maths: How to Move a Shape Using Column Vectors

Translation is the first transformation most students meet, and it is the one that causes the fewest headaches once the notation clicks. The idea itself is simple: slide a shape from one place to another without turning it, flipping it or changing its size.

The marks in an exam are usually lost not on the sliding but on the description. This guide covers the rule, the notation, the steps, two worked examples and the mistakes that cost the most marks.

What is a translation?

A translation moves every point of a shape the same distance in the same direction. Nothing else about the shape changes.

The starting shape is called the object. The shape after the move is called the image. Because only the position changes, the object and the image are congruent, which means identical in shape and size.

Properties of a translation

•      The side lengths stay the same

•      The angles stay the same

•      The area and perimeter stay the same

•      The orientation stays the same, so the shape is never flipped or rotated

•      Every vertex moves by exactly the same amount

If the shape has turned or flipped, it is not a translation. That is the quickest check you can make in an exam.

Column vector notation

A translation is described using a column vector, written as two numbers stacked inside a single bracket. Written on one line for convenience, a column vector (a, b) means:

•      a is the horizontal movement. A positive value moves right, a negative value moves left.

•      b is the vertical movement. A positive value moves up, a negative value moves down.

Read the top number first, then the bottom number. The order is the same as coordinates, across before up, which is why the phrase along the corridor and up the stairs is still the most reliable memory aid in the room.

Column vector	Meaning
(5, 0)	5 right, no vertical movement
(0, 3)	No horizontal movement, 3 up
(4, 2)	4 right and 2 up
(-3, 6)	3 left and 6 up
(2, -7)	2 right and 7 down
(-4, -1)	4 left and 1 down

How to translate a shape, step by step

1.      Read the column vector carefully and note the direction of each number.

2.      Label the vertices of the object, for example A, B and C.

3.      Take the first vertex and count the horizontal movement first, then the vertical movement. Mark the new point.

4.      Repeat for every remaining vertex. Do not estimate the rest by eye.

5.      Join the new points in the same order as the original shape.

6.      Label the image using primes, so A becomes A prime, B becomes B prime and so on.

7.      Check that the image is the same size and the same way round as the object.

Translating every vertex individually is slower but it is the method that survives pressure. Sliding the whole shape by eye is where most careless errors come from.

Worked example 1

Triangle ABC has vertices at A(1, 2), B(4, 2) and C(1, 5). Translate the triangle by the vector (3, -1).

The vector means 3 right and 1 down. Add 3 to every x coordinate and subtract 1 from every y coordinate.

Vertex	Original	Working	Image
A	(1, 2)	(1 + 3, 2 - 1)	(4, 1)
B	(4, 2)	(4 + 3, 2 - 1)	(7, 1)
C	(1, 5)	(1 + 3, 5 - 1)	(4, 4)

The image is the triangle with vertices A prime (4, 1), B prime (7, 1) and C prime (4, 4). Notice that the shape is still a right angled triangle with the same side lengths, which confirms the work.

Worked example 2: describing a translation

Shape P has a vertex at (6, 3). After a translation, the matching vertex on shape Q is at (2, 8). Describe the translation fully.

Subtract the object coordinate from the image coordinate, always image minus object.

•      Horizontal: 2 minus 6 gives negative 4, so the shape moved 4 left

•      Vertical: 8 minus 3 gives positive 5, so the shape moved 5 up

The full answer is: a translation by the vector (-4, 5).

To get full marks you must use the word translation and give the column vector. Writing only 4 left and 5 up, or giving the vector without naming the transformation, usually loses a mark.

Translation compared with the other transformations

Transformation	What changes	What you must state
Translation	Position only	The word translation and a column vector
Reflection	Position and orientation, mirrored	The word reflection and the equation of the mirror line
Rotation	Position and orientation, turned	Angle, direction and centre of rotation
Enlargement	Size, and position	Scale factor and centre of enlargement

Translation, reflection and rotation all preserve size, so the object and image remain congruent. Enlargement is the only one of the four that changes the size, producing similar rather than congruent shapes.

Common mistakes to avoid

•      Reading the column vector upside down. The top number is always horizontal. Reversing the two is the single most common error.

•      Getting the sign wrong. Negative on top is left, negative on the bottom is down.

•      Calculating object minus image instead of image minus object when describing a translation, which reverses both signs.

•      Moving only one vertex and drawing the rest by eye.

•      Forgetting to write the word translation in a describe question.

•      Giving the answer as coordinates, for example (-4, 5) written as a point rather than as a vector.

Practice questions

8.      Translate the point (2, 7) by the vector (5, -3).

9.      Translate the point (-1, 4) by the vector (-2, -6).

10.   Square WXYZ has a vertex W at (0, 0). After a translation the matching vertex W prime is at (3, 4). Describe the translation fully.

11.   A shape is translated by (7, 2) and then by (-7, -2). Describe the single transformation that has the same effect.

12.   Triangle DEF has vertices D(2, 1), E(5, 1) and F(2, 3). Translate it by (-4, 2) and give the coordinates of the image.

Answers

13.   (7, 4)

14.   (-3, -2)

15.   A translation by the vector (3, 4)

16.   The shape returns to its starting position, so the combined effect is a translation by the vector (0, 0), which leaves the shape unchanged

17.   D prime (-2, 3), E prime (1, 3), F prime (-2, 5)

Frequently Asked Questions

Q. Does a translation change the size of a shape?

A. No. A translation only changes position. The side lengths, angles, perimeter and area all stay exactly the same, so the object and the image are congruent.

Q. Which number in a column vector comes first?

A. The top number always describes horizontal movement, positive for right and negative for left. The bottom number describes vertical movement, positive for up and negative for down.

Q. How do I describe a translation in an exam?

A. Write the word translation and give the column vector. Both parts are needed. Work out the vector by subtracting the object coordinate from the matching image coordinate.

Q. Can a translation be diagonal?

A. Yes. When both numbers in the vector are non zero, the shape moves diagonally. It still counts as a single translation, not two separate ones.

Q. What is the difference between translation and rotation?

A. A translation slides the shape without turning it, so the orientation is unchanged. A rotation turns the shape about a fixed point, so the orientation changes even though the size does not.

Disclaimer

This article is for general educational guidance. Always follow the notation and marking requirements set by your own exam board and teacher.