# How to Translate a Shape Using Cartesian Coordinates

## Translating a Shape Using Cartesian Coordinates

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.

We have translated the shape. ## Slider

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

Open the slider in a new tab