Translations intro

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 segment by dragging it from the middle, not the endpoints:

Notice how the segment's direction and length stayed the same as you moved it. Translations only move things from one place to another; they don't change their size, 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.

Coordinates allow us to be very precise about the translations we perform.

Without coordinates, we could say something like, "We get $\maroonD{B'}$start color #ca337c, B, prime, end color #ca337c by translating $\blueD{B}$start color #11accd, B, end color #11accd 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 $\maroonD{B'}$start color #ca337c, B, prime, end color #ca337c by translating $\blueD{B}$start color #11accd, B, end color #11accd by 5 units to the right and 4 units down."

More compactly, we can describe this as a translation by $\langle 5,-4 \rangle$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.

For any transformation, we have the pre-image 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 pre-image point was $\blueD{B}$start color #11accd, B, end color #11accd and the image point was $\maroonD{B'}$start color #ca337c, B, prime, end color #ca337c.

Note that we indicated the image by $\maroonD{B'}$start color #ca337c, B, prime, end color #ca337c, pronounced B prime. It is common, when working with transformations, to use the same letter for the image and the pre-image, simply adding the "prime" suffix to the image.