What is translation?
In geometry, the word translation means moving.
It can help to think of translating a shape as sliding the shape.
When you translate a shape:
- every point on the shape moves the same distance and in the same direction.
- the size of the shape stays the same.
- the shape is not rotated - its orientation stays the same.
The original shape and the resulting images are congruent, which means they have the same shape and size.
Activity: Translating in four quadrants
Discover how to translate a shape in this interactive activity and then put your knowledge to the test with a quiz.
How to translate shapes in four quadrants
In geometry, translating a shape means moving it without changing its size, shape or orientation.
You can translate a shape on a grid by sliding it to a new position.
You can move the shape left or right on the x-axis and up and down on the y-axis.
If the shape is in a four quadrant grid like this one, there might be negative coordinates as well.
On this grid, the position of the blue triangle has been translated to a new position to make the pink triangle.
The blue triangle's top corner marked point A starts at (-5, 5). It moves 7 units right and then 6 units down, it ends up at (2, -1) as point A1 on the pink triangle.
You do not need to count on the grid to work out new coordinates. You can use your understanding of the x- and y-axis to find out the new coordinates.
Moving a shape left or right changes the x-coordinates, while moving it up or down changes the y-coordinates.
For example, if a point at (4, 3) is moved 4 units left, you subtract 4 from the x-coordinate, making it 0.
If it moved 4 units right, you add 4 to the x-coordinate, make it 8.
If the same point is then moved 2 units up, you add 2 to the y-coordinate, making it 5.
Or if you moved it 2 units down, you subtract 2 from the y-co-ordinate, making it 1.
Example 1
Take a look at the two triangles on this grid.
The right-angled triangle ABC is positioned in the second quadrant.
The coordinates of the triangle are:
| Point | Coordinates |
|---|---|
| A | -4, 5) |
| B | (-1, 2) |
| C | -4, 2) |
The triangle ABC is translated 3 units to the right (parallel to the x-axis) and 6 units down (parallel to the y-axis)
What are the coordinates of the new triangle A¹B¹C¹?
✓ The resulting image of this translation has been labeled A¹B¹C¹.
This means the new coordinates are:
| Point | Coordinate |
|---|---|
| A¹ | (-1, -1) |
| B¹ | (2, -4) |
| C¹ | (-1, -4) |
The triangle ABC and the triangle A¹B¹C¹ are congruent.
Congruent means that the two shapes have the same size and the same shape.
Example 2
Now, here are the points of a parallelogram plotted on a grid.
The coordinates of the parallelogram are:
| Point | Coordinates |
|---|---|
| A | (-7, -5) |
| B | (-3, -5) |
| C | (-1, -8) |
| D | -5, -8) |
The parallelogram is translated 3 units to the left (parallel to the x-axis) and 4 units up (parallel to the y-axis).
Here are the new coordinates for three of the points:
| Point | Coordinates |
|---|---|
| A¹ | (-10, -1) |
| B¹ | (-6, -1)) |
| C¹ | (-4, -4) |
| D¹ | ? |
What will be the new coordinate for point D?
✓ Let's work it out step by step.
The starting coordinates for D are (-5, -8).
First let's translate 3 units to the left.
Move the coordinate 3 units to the left on the grid or subtract 3 from the x-coordinate.
-5 - 3 = -8
Now let's translate 4 units up.
Move the coordinate 4 units up on the grid or add 4 to the y-coordinate.
-8 + 4 = -4
So, the new coordinates for point D are D¹ (-8, -4).
Example 3
Here are two trapeziums plotted on a grid.
The trapezium ABCD is positioned in the second quadrant.
The coordinates of the trapezium are:
| Point | Coordinates |
|---|---|
| A | (-5, 4) |
| B | (-3, 4) |
| C | (-2, 1) |
| D | (-6, 1) |
After a translation, the new coordinates are:
| Point | Coordinates |
|---|---|
| A¹ | (2, -1) |
| B¹ | (4, -1) |
| C¹ | (5, -4) |
| D¹ | (1, -4) |
But, how has it been translated?
✓ The new coordinates of D¹ are (1, -4).
Let's look at the x coordinate first.
It went from -6 to 1. Look on the grid, which direction has it moved in?
It's moved 7 units to the right.
Now let's look at the y coordinate.
It went from 1 to -4. Look on the grid, which direction is that?
It's moved down by 5 units.
You can check this with other points to make sure.
The trapezium ABCD was translated 7 units to the right and 5 units down.
Play Guardians: Defenders of Mathematica to get ready for SATs. game
In this game, use the times tables and more maths skills to defeat monsters and reclaim the Kingdom.
More on Coordinates
Find out more by working through a topic
- count1 of 4
- count2 of 4
- count3 of 4
