# 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:

### Din tur!

#### Problem 1

#### Problem 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!

#### Problem 1

#### Problem 2

#### Challenge problem

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