Learn what translations are and how to perform them in our interactive widget.
To see what a translation is, please grab the point and move it around.
Nice! You translated the point. In geometry, a translation moves a thing up and down or left and right.
Here, try translating this line:
Notice how the line's disposition remained the same as you moved it. Translations only move things from one place to another; they don't change their arrangement or direction.
Now that we've got a basic understanding of what translations are, let's learn how to use them on the coordinate plane.

Translations on the coordinate plane

Coordinates allow us to be very precise about the translations we perform.
Without coordinates, we could say something like, "We get start color maroonD, B, prime, end color maroonD by translating start color blueD, B, end color blueD down and to the right."
But that's not very precise. If we use a coordinate grid, we can say something more exact: "We get start color maroonD, B, prime, end color maroonD by translating start color blueD, B, end color blueD by 5 units to the right and 4 units down."
More compactly, we can describe this as a translation by open angle, 5, comma, minus, 4, close angle.
The negative sign in front of the 4 tells us the vertical shift is downwards instead of upwards. Similarly, a translation to the left is indicated by the first value being negative.

Sources and images

For any transformation, we have the source figure, which is the figure we are performing the transformation upon, and the image figure, which is the result of the transformation. For example, in our translation, the source point was start color blueD, B, end color blueD and the image point was start color maroonD, B, prime, end color maroonD.
Note that we indicated the image by start color maroonD, B, prime, end color maroonD, pronounced B prime. It is common, when working with transformations, to use the same letter for the image and the source, simply adding the "prime" suffix to the image.

Let's try some practice problems

Problem 1

Translate the line segment by open angle, minus, 8, comma, 3, close angle.

First, click on the Translate button. The segment's endpoints will light up.
Then, drag one of the endpoints 8 units to the left and 3 units up.
Once you start moving the segment, a label will indicate the translation according to the current position of the segment.

Problem 2

Translate the triangle by open angle, 2, comma, 6, close angle.

First, click on the Translate button. The triangle's vertices will light up.
Then, drag one of the vertices 2 units to the right and 6 units up.
Once you start moving the triangle, a label will indicate the translation according to the current position of the triangle.

Problem 3

Translate the circle by open angle, minus, 8, comma, minus, 1, close angle.

First, click on the Translate button. The circle's center will light up.
Then, drag the center of the circle 8 units to the left and 1 unit down.
Once you start moving the circle, a label will indicate the translation according to the current position of the circle.

Challenge problem 1

Draw the image of the line segment under a translation by open angle, 2, comma, minus, 3, close angle.

A translation by open angle, 2, comma, minus, 3, close angle means the line segment should move 2 units to the right and 3 units down.
Since we are supposed to draw the image by dragging the endpoints of the segment, let's move each endpoint 2 units to the right and 3 units down.

Challenge problem 2

What translation maps point start color blueD, C, end color blueD to point start color maroonD, C, prime, end color maroonD?
Choose 1 answer:
Choose 1 answer:

To get from start color blueD, C, end color blueD to start color maroonD, C, prime, end color maroonD, we must go 1 unit left and 4 units up.
Therefore, start color maroonD, C, prime, end color maroonD is obtained through a translation of start color blueD, C, end color blueD by open angle, minus, 1, comma, 4, close angle.