The Lesson

A translation moves a shape. Every point of the shape is moved in the same direction by the same distance.

Using Cartesian Coordinates to Translate a Shape

Cartesian coordinates can be used to translate a shape.

Cartesian Coordinates Tell You Where a Point Is

Cartesian coordinates are used to describe the position of a point on a graph. They consist of a pair of coordinates: The image below shows the Cartesian coordinate of a point on a shape.In this image, the x-coordinate of the point is 1 and the y-coordinate of the point is 2. The Cartesian coordinates are (1, 2).

A Column Vector Tells You How to Move Each Point

A column vector tells us how to translate a shape. A column vector shows two numbers, one above the other. An example of a column vector is shown below:
  • The top number (4) tells us how much to move the shape across.
  • The bottom number (2) tells us how much to move the shape up.

Using the Column Vector to Find the Cartesian Coordinates of the Translated Shape

The numbers in the column vector are added to the Cartesian coordinates of the point.
  • The top number of the column vector is added to the x-coordinate of the point. The image below shows the top number of the column vector (4) being added to the x-coordinate of the point (1):

    The point on the translated shape is 5 (= 1 + 4)
  • The bottom number of the column vector is added to the y-coordinate of the point. The image below shows the bottom number of the column vector (2) being added to the y-coordinate of the point (2):

    The point on the translated shape is 4 (= 2 + 2)
The Cartesian coordinates of the translated point is (5, 4).

How to Translate a Shape Using Cartesian Coordinates

Translating a shape using Cartesian coordinates is easy.

Question

Translate the shape below using the column vector.

Step-by-Step:

Let us start by choosing a point on the shape and translating it. We will translate point A.

1

Find the Cartesian coordinates of the point.

In our example, the Cartesian coordinates the point A is (1, 3).

2

Add the top number of the column vector (4) to the x-coordinate of the point (1).

1 + 4 = 5
The x-coordinate of the translated point is 5.

3

Add the bottom number of the column vector (2) to the y-coordinate of the point (3).

3 + 2 = 5
The y-coordinate of the translated point is 5.

4

Draw the translated point at with the x-coordinate found in Step 2 (5) and the y-coordinate found at Step 3 (5). The translated point has Cartesian coordinates of (5, 5).

Point A has been translated to to find A', the corresponding point on the translated shape.
Repeat the Steps for Points B and C. It can be easier to translate each point in a table:
  • Column (1) has the x-coordinate of each point.
  • Column (2) has the y-coordinate of each point.
  • Column (3) adds the top number in the column vector (4) to the number in column (1).
  • Column (4) adds the bottom number in the column vector (2) to the number in column (2).
(1) (2) (3) = (1) + 4 (4) = (2) + 2
Point X-Coordinate Y-Coordinate Translated X-Coordinate Translated Y-Coordinate Point
A 1 3 5 5 A'
B 1 2 5 4 B'
C 2 2 6 4 C'
Plot these points and draw in the shape.

Answer:

We have translated the shape.

Lesson Slides

The slider below shows another real example of how to translate a shape using Cartesian coordinates.

What Is a Translation?

A translation is a slide of a shape: without rotating, reflecting or resizing it. A translation is a type of transformation.