Translation in a Direction

A translation is the movement of an object from one place to another on the 2D coordinate plane. The shape of the object does not change after it is translated; it is just moved to a different place.

(Video) Translation

by Planet Nutshell

When I hear the word translation, the first thing I think of is the translation between languages. Ideally, the translation of a sentence from say, English to Japanese, should not change the meaning of it. In reality, the differences between languages and cultures may alter the meaning, or the way we understand it.

What about translation in a geometric sense? On a dinner table, if you slide down a hot pot of soup, does anything change about the soup or the pot?

Here is a video explaining what translation is.

This video explains what translation means in geometry. It also talks about how to notate a translation in a coordinate plane. Here is a problem to test you understanding of translation.

(Video) How Do I Translate a Figure?

by MashUp Math

After finding out what translation is, now let's learn about some properties of translation. Here is a helpful video.

Here is a brief summary of what happened in the video: $\\$

A translation is NOT a change in size or orientation

Notation $\\$ We denote a translation as $T_{a,b}$. Applying this translation to a point $P(X, \,Y)$, we get $P'(X', \,Y')$. Here is an equation expressing the relationship between the two points: $$P'(X', \,Y') = P(X+a, \,Y+b)$$

Try the following problem to test your understanding.

A translation is when you "slide" an object from one place on the 2D plane to another. The translated object stays congruent and it stays in the same orientation (which is changed by rotation). All angles and side lengths stay the same.

For reference, here is "What is a congruence transformation?"

Just to review:

Isometries are a type of transformation, and if you read the title you might have guessed that a translation is a type of isometry.

Analogy

So now we have an intuitive understand of what is an isometry. Even though the technical definition is "distances are preserved," we can think of isometries as transformations that preserve shape.

Diagrams and Examples

Now we will establish what it means when we say "A translation is an isometry"

Consider a triangle $ABC.$

Remember that we can think of isometries as transformations that preserve shape.

Great job! This is exactly the idea behind why translations are isometries.

Translation, in geometry, simply means "moving" a shape without actually rotating or changing the size of it. All the points of that particular shape must move the same distance and more importantly, the same direction. There are quite a few ways of translating a shape. However, the easiest and the MOST effective way is the x-and-y units method.