Transformations of Shapes

Think of a translation as picking a shape up and sliding it across the page without turning it, flipping it, or resizing it at all. Imagine sliding a book across a table — it stays exactly the same book, just in a new spot. That's a translation. Because nothing about the shape itself changes, the object and the image are congruent (identical in size and shape) — the only thing that's different is where it's sitting.

The translation vector

Every translation is described using a column vector, written like this:

So a vector of (3, −1) means "3 squares right, 1 square down." A vector of (−4, 5) means "4 squares left, 5 squares up." The easiest way to avoid mixing up left/right and up/down is to translate one vertex first, mark it with a cross, then work around the shape vertex by vertex, copying the same movement each time. Don't try to shift the whole shape in one go by eye — you'll make mistakes.

Describing a translation (the reverse process)

Sometimes you're given the object and the image and asked to find the vector. Here's exactly how:

1. Pick any vertex on the object (say, a corner labelled A).
2. Find the corresponding vertex on the image (A′).
3. Count how far left/right you moved — that's your x value (negative if you went left).
4. Count how far up/down you moved — that's your y value (negative if you went down).

Reversing a translation

If a shape has been translated by (x, y), to bring it back to where it started you simply flip the sign of both numbers.