### Translations

A **transformation** is an operation that moves, flips, or otherwise changes a figure to create a new figure. A **rigid transformation** (also known as an **isometry** or **congruence transformation**) is a transformation that does not change the size or shape of a figure.

The rigid transformations are **translations**, reflections, and rotations. The new figure created by a transformation is called the **image**. The original figure is called the **preimage**. If the preimage is \begin{align*}A\end{align*}, then the image would be \begin{align*}A'\end{align*}, said “a prime.” If there is an image of \begin{align*}A'\end{align*}, that would be labeled \begin{align*}A''\end{align*}, said “a double prime.”

A **translation** is a transformation that moves every point in a figure the same distance in the same direction. For example, this transformation moves the parallelogram to the right 5 units and up 3 units. It is written \begin{align*}(x,y) \rightarrow (x+5,y+3)\end{align*}.

What if you were given the coordinates of a quadrilateral and you were asked to move that quadrilateral 3 units to the left and 2 units down? What would its new coordinates be?

### Examples

#### Example 1

Triangle \begin{align*}\triangle ABC\end{align*} has coordinates \begin{align*}A(3, -1), B(7, -5)\end{align*} and \begin{align*}C(-2, -2)\end{align*}. Translate \begin{align*}\triangle ABC\end{align*} to the left 4 units and up 5 units. Determine the coordinates of \begin{align*}\triangle A'B'C'\end{align*}.

Graph \begin{align*}\triangle ABC\end{align*}. To translate \begin{align*}\triangle ABC\end{align*}, subtract 4 from each \begin{align*}x\end{align*} value and add 5 to each \begin{align*}y\end{align*} value of its coordinates.

\begin{align*}& A(3,-1) \rightarrow (3-4,-1+5)=A'(-1,4)\\ & B(7,-5) \rightarrow (7-4,-5+5)=B'(3,0)\\ & C(-2,-2) \rightarrow (-2-4,-2+5)=C'(-6,3)\end{align*}

The rule would be \begin{align*}(x,y) \rightarrow (x-4,y+5)\end{align*}.

#### Example 2

Using the translation \begin{align*}(x,y) \rightarrow (x+2,y-5)\end{align*}, what is the image of \begin{align*}A(-6, 3)\end{align*}?

\begin{align*}A' (-4, -2)\end{align*}

#### Example 3

Graph square \begin{align*}S(1, 2), Q(4, 1), R(5, 4)\end{align*} and \begin{align*}E(2, 5)\end{align*}. Find the image after the translation \begin{align*}(x,y) \rightarrow (x-2,y+3)\end{align*}. Then, graph and label the image.

We are going to move the square to the left 2 and up 3.

\begin{align*}(x,y) &\rightarrow (x-2,y+3)\\ S(1,2) &\rightarrow S'(-1,5)\\ Q(4,1) &\rightarrow Q'(2,4)\\ R(5,4) &\rightarrow R'(3,7)\\ E(2,5) &\rightarrow E'(0,8)\end{align*}

#### Example 4

Find the translation rule for \begin{align*}\triangle TRI\end{align*} to \begin{align*}\triangle T'R'I'\end{align*}.

Look at the movement from \begin{align*}T\end{align*} to \begin{align*}T'\end{align*}. The translation rule is \begin{align*}(x,y) \rightarrow (x+6,y-4)\end{align*}.

### Review

Use the translation \begin{align*}(x,y) \rightarrow (x+5,y-9)\end{align*} for questions 1-7.

- What is the image of \begin{align*}A(-1, 3)\end{align*}?
- What is the image of \begin{align*}B(2, 5)\end{align*}?
- What is the image of \begin{align*}C(4, -2)\end{align*}?
- What is the image of \begin{align*}A'\end{align*}?
- What is the preimage of \begin{align*}D'(12, 7)\end{align*}?
- What is the image of \begin{align*}A''\end{align*}?
- Plot \begin{align*}A, A', A'',\end{align*} and \begin{align*}A'''\end{align*} from the questions above. What do you notice?

The vertices of \begin{align*}\triangle ABC\end{align*} are \begin{align*}A(-6, -7), B(-3, -10)\end{align*} and \begin{align*}C(-5, 2)\end{align*}. Find the vertices of \begin{align*}\triangle A'B'C'\end{align*}, given the translation rules below.

- \begin{align*}(x,y) \rightarrow (x-2,y-7)\end{align*}
- \begin{align*}(x,y) \rightarrow (x+11,y+4)\end{align*}
- \begin{align*}(x,y) \rightarrow (x,y-3)\end{align*}
- \begin{align*}(x,y) \rightarrow (x-5,y+8)\end{align*}
- \begin{align*}(x,y) \rightarrow (x+1,y)\end{align*}
- \begin{align*}(x,y) \rightarrow (x+3,y+10)\end{align*}

In questions 14-17, \begin{align*}\triangle A'B'C'\end{align*} is the image of \begin{align*}\triangle ABC\end{align*}. Write the translation rule.

Use the triangles from #17 to answer questions 18-20.

- Find the lengths of all the sides of \begin{align*}\triangle ABC\end{align*}.
- Find the lengths of all the sides of \begin{align*}\triangle A'B'C'\end{align*}.
- What can you say about \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle A'B'C'\end{align*}? Can you say this for
*any*translation? - If \begin{align*}\triangle A'B'C'\end{align*} was the
*preimage*and \begin{align*}\triangle ABC\end{align*} was the image, write the translation rule for #14. - If \begin{align*}\triangle A'B'C'\end{align*} was the
*preimage*and \begin{align*}\triangle ABC\end{align*} was the image, write the translation rule for #15. - Find the translation rule that would move \begin{align*}A\end{align*} to \begin{align*}A'(0, 0)\end{align*}, for #16.
- The coordinates of \begin{align*}\triangle DEF\end{align*} are \begin{align*}D(4, -2), E(7, -4)\end{align*} and \begin{align*}F(5, 3)\end{align*}. Translate \begin{align*}\triangle DEF\end{align*} to the right 5 units and up 11 units. Write the translation rule.
- The coordinates of quadrilateral \begin{align*}QUAD\end{align*} are \begin{align*}Q(-6, 1), U(-3, 7), A(4, -2)\end{align*} and \begin{align*}D(1, -8)\end{align*}. Translate \begin{align*}QUAD\end{align*} to the left 3 units and down 7 units. Write the translation rule.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.3.