# Translating shapes

Learn how to draw the image of a given shape under a given translation.

## Introduction

In this article, we'll practice the art of translating shapes. Mathematically speaking, we will learn how to draw the image of a given shape under a given translation.
A translation by $\langle a,b \rangle$ is a transformation that moves all points $a$ units in the $x$-direction and $b$ units in the $y$-direction. Such a transformation is commonly represented as $T_{(a,b)}$.

## Part 1: Translating points

### Let's study an example problem

Find the image $A'$ of $A(4,-7)$ under the transformation $T_{(-10,5)}$.

### Løsning

The translation $T_{(\tealD{-10},\maroonD{5})}$ moves all points $\tealD{-10}$ in the $x$-direction and $\maroonD{+5}$ in the $y$-direction. In other words, it moves everything 10 units to the left and 5 units up.
Now we can simply go 10 units to the left and 5 units up from $A(4,-7)$.
We can also find $A'$ algebraically:
$A'=(4\tealD{-10},-7\maroonD{+5})=(-6,-2)$

## Part 2: Translating line segments

### Let's study an example problem

Consider line segment $\overline{CD}$ drawn below. Let's draw its image under the translation $T_{(9,-5)}$.

### Løsning

When we translate a line segment, we are actually translating all the individual points that make up that segment.
Luckily, we don't have to translate all the points, which are infinite! Instead, we can consider the endpoints of the segment.
Since all points move in exactly the same direction, the image of $\overline{CD}$ will simply be the line segment whose endpoints are $C'$ and $D'$.

## Part 3: Translating polygons

### Let's study an example problem

Consider quadrilateral $EFGH$ drawn below. Let's draw its image, $E'F'G'H'$, under the translation $T_{(-6,-10)}$.

### Løsning

When we translate a polygon, we are actually translating all the individual line segments that make up that polygon!
Basically, what we did here is to find the images of $E$, $F$, $G$, and $H$ and connect those image vertices.

### Din tur!

#### Challenge problem

The translation $T_{(4,-7)}$ mapped $\triangle PQR$. The image, $\triangle P'Q'R'$, is drawn below.