Math Core

Slide a book across a table and it ends up somewhere new, but it is still the same book: same size, same shape, still facing the same way. In geometry, that kind of slide is called a translation, and it is the simplest of the moves you will study in this unit.

Pre-images and images

A transformation is a rule that moves every point of a figure to a new location. The original figure is the pre-image, and the figure you get after the move is the image.

To keep track of which point went where, the image of a point gets the same letter with a prime mark. The image of point $A$ is $A'$ (read "A prime"), the image of $B$ is $B'$ , and so on.

Definition

Translation

A translation slides every point of a figure the same distance in the same direction. On the coordinate plane, a translation $a$ units horizontally and $b$ units vertically follows the rule

(x, y) \rightarrow (x + a,\ y + b)

A positive $a$ moves right and a negative $a$ moves left. A positive $b$ moves up and a negative $b$ moves down.

Translating a figure

To translate a polygon, translate each vertex, then connect the new vertices in the same order.

Worked example: Translating a triangle

Translate triangle $ABC$ with $A(1, 1)$ , $B(4, 1)$ and $C(2, 4)$ 5 units left and 3 units down.

Left 5 means subtract 5 from each x-coordinate. Down 3 means subtract 3 from each y-coordinate. The rule is $(x, y) \rightarrow (x - 5,\ y - 3)$ .

\begin{aligned} A(1, 1) &\rightarrow A'(1 - 5,\ 1 - 3) = A'(-4, -2) \\ B(4, 1) &\rightarrow B'(4 - 5,\ 1 - 3) = B'(-1, -2) \\ C(2, 4) &\rightarrow C'(2 - 5,\ 4 - 3) = C'(-3, 1) \end{aligned}

Triangle ABC (pre-image) slides 5 left and 3 down to triangle A′B′C′ (image).Open in grapher →

Every vertex moved exactly the same way, so the whole triangle slid without turning or flipping.

Common mistake

Left and right change the x-coordinate. Up and down change the y-coordinate. A common slip is subtracting from the wrong coordinate. Say the direction out loud ("left 5, so $x - 5$ ") before you calculate.

Finding the rule

If you know one point and its image, you can find the translation by comparing coordinates.

Worked example: Working out the translation

A translation moves $P(-2, 3)$ to $P'(4, 1)$ . Write the rule, and find the image of $Q(0, -5)$ .

The x-coordinate went from $-2$ to $4$ , a change of $4 - (-2) = 6$ . The y-coordinate went from $3$ to $1$ , a change of $1 - 3 = -2$ . The translation is 6 right and 2 down:

(x, y) \rightarrow (x + 6,\ y - 2)

Apply it to $Q$ : $Q'(0 + 6,\ -5 - 2) = Q'(6, -7)$ .

Worked example: Working backward

After the translation $(x, y) \rightarrow (x - 2,\ y + 7)$ , the image of point $D$ is $D'(3, -4)$ . Where was $D$ ?

Undo each step. The rule subtracted 2 from $x$ , so add 2 back: $3 + 2 = 5$ . The rule added 7 to $y$ , so subtract 7: $-4 - 7 = -11$ .

$D$ was at $(5, -11)$ . Check: $(5 - 2,\ -11 + 7) = (3, -4)$ . ✓

What stays the same

A translation is a rigid motion: it moves a figure without stretching, shrinking or bending it.

Translations preserve

Lengths: each side of the image is as long as the matching side of the pre-image.
Angle measures: each angle keeps its size.
Parallel lines: sides that were parallel stay parallel.
Orientation: the figure faces the same way. If you read $A \to B \to C$ clockwise on the pre-image, $A' \to B' \to C'$ is also clockwise.

So the image is always an exact copy of the pre-image.

Tip

To check a translation on a graph, draw an arrow from each vertex to its image. The arrows should all be the same length and point in the same direction.

Practice

Practice 1

Translate the point $(3, -2)$ 4 units right and 6 units up. What is the image?

Enter a point like (2, -3)

Practice 2

Use the rule $(x, y) \rightarrow (x - 5,\ y + 2)$ to find the image of $(-1, -4)$ .

Enter a point like (2, -3)

Triangle ABC and its image A′B′C′.Open in grapher →

Practice 3

Which rule describes the translation in the graph above?

Practice 4

A triangle has a perimeter of 24 cm. It is translated 10 units right. What is the perimeter of the image, in centimeters?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Point $M$ is translated 3 units left and 8 units down. Its image is $M'(2, -6)$ . What are the coordinates of $M$ ?

Enter a point like (2, -3)

Practice 6

A translation moves $A(-3, 5)$ to $A'(1, 2)$ . The same translation moves $B(6, -1)$ to $B'$ . What are the coordinates of $B'$ ?

Enter a point like (2, -3)

Practice 7

The point $(4, 1)$ is translated 2 units right and 5 units down. Then its image is translated 7 units left and 1 unit up. Where does the point end up?

Enter a point like (2, -3)