# Determining translations

Learn how to find the necessary translation to map a given source shape onto a given image shape.
In this article, we will solve problems where we are given starting and ending coordinates and asked to figure out what translation must have occurred.

# Part 1: Determining the translation for a single pair of points

## Let's study an example problem

A translation maps point $A(3,7)$ to point $A'(6,-2)$. Let's determine what translation this is.

### Løsning

Step 1: Horizontal shift. $A$ is shifted $3$ units to the right because $(6)-(3)=\tealD{+3}$.
Step 2: Vertical shift. $A$ is shifted $9$ units down because $(-2)-(7)=\maroonD{-9}$.
The answer: $A$ is mapped onto $A'$ under a translation by $\langle \tealD{3},\maroonD{-9} \rangle$.

## Challenge problem

A certain translation takes point $D(-3,10)$ to point $D'(-12,21)$.

# Part 2: Determining the translation for a pair of polygons

## Let's study an example problem

Consider the quadrilaterals drawn below. Let's determine the translation that maps the source $\blueD{FGHI}$ onto the image $\maroonD{F'G'H'I'}$.

### Løsning

Let's focus in on a pair of corresponding points, such as $F(-4,6)$ and $F'(2,3)$. If we can find the translation that takes $F$ to $F'$, we will necessarily know the translation that takes the entire source quadrilateral to its image!
Horizontal shift: $(2)-(-4)=\tealD{+6}$
Vertical shift: $(3)-(6)=\maroonD{-3}$
Therefore, $FGHI$ is mapped onto $F'G'H'I'$ under a translation by $\langle\tealD{6},\maroonD{-3}\rangle$.