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### Course: Congruence, Similarity, Right Triangles, & Trig 231-240>Unit 2

Lesson 2: Translations

# Translations intro

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 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.

## 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 ${B}^{\prime }$ by translating $B$ 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 ${B}^{\prime }$ by translating $B$ by 5 units to the right and 4 units down."
More compactly, we can describe this as a translation by $⟨5,-4⟩$.
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.

### Pre-images and images

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 $B$ and the image point was ${B}^{\prime }$.
Note that we indicated the image by ${B}^{\prime }$, 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.

## Let's try some practice problems

### Problem 1

Each unit in the grid equals $1$.
Draw the image of the line segment under a translation by $⟨2,-3⟩$.

### Problem 2

Each unit in the grid equals $1$.
Draw the image of the circle after a translation by $⟨-7,-1⟩$.

### Challenge problem

What translation maps point $C$ to point ${C}^{\prime }$?

## Want to join the conversation?

• What can I do to relate to this in my life
• Moving. Anything.
If you walk to your door, you're technically translating yourself from where you are to the door, whilst it's in 3D you can still think of walking maybe North 1 meter and West 3 meters, or you could be walking to the store, you go from your house to the store a certain distance one way, then more distance another way which will end up with you in the position of the store. Anytime something moves from one point to another, that's a translation
• im confused when it doesnt tell you to expand the circle
• its bc the website probably glitched out or they just forgot to make the circle the size it was supposed to be
• How is this going to help me get a job
• Engineering
• For challenge question 2, how come it isn't the first answer?
• Because if you moved it (1,4), it would end C" would end up 2 spaces to the right, as a movement of (1,4) from point C means the same thing as moving point C 1 space to the right, and four spaces up. The correct answer, answer c, moves point C (-1,4)- 1 space to the left (-1), and 4 spaces up (4). Hope this helps.
• Why is translation, rotation, and reflection needed in real life.
• Engineering
• What are the types of translations? Other than left and right and up and down.
• Hi Aidan,

Translations can make an object move only left, only right, only up, only down or a combination of them, such as left & up, left & down.

I hope this helps.

Aiena.
• How is it -1, 4. It's moving to the right once and moving down 4. Shouldn't the transformation be 1, -4?
• It is a challenge problem because you are moving from C to C', you show movement from C' to C. The prime is always the image, and the non-prime is the preimage.