Translations

A translation slides every point of a figure the same distance in the same direction. On a coordinate grid, the rule (x, y) \to (x + a, y + b) moves each point a units left or right and b units up or down. The image has the same size and shape as the original figure.

Before this lesson: You should know the Coordinate Plane, Adding and Subtracting Integers and the Distance Formula.

What Is a Translation?

Picture a book on a desk. Push it across the desk without turning it. Every corner of the book moves the same way. That kind of move is a translation, or a slide.

A translation is one kind of transformation. A transformation is a rule that moves or changes a figure. The figure you start with is the pre-image. The figure you end with is the image.

We name image points with a prime mark. Point A moves to point A', read “A prime.” So triangle ABC moves to triangle A'B'C'.

A translation is a rigid motion. That means it keeps every length and every angle the same. The image is congruent to the pre-image. It also faces the same way, because nothing turns or flips.

Translation Rule

On a coordinate grid, a translation adds the same number to every x-coordinate. It also adds the same number to every y-coordinate.

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

The number a moves points left or right. The number b moves points up or down. You can also write the move as a translation vector, \langle a, b \rangle. A vector is an arrow with a length and a direction.

Right and up use plus signs. Left and down use minus signs.

How to Translate a Figure

1. Find a and b from the rule, the vector or the words.
2. Add a to the x-coordinate of each vertex.
3. Add b to the y-coordinate of each vertex.
4. Plot the new points. Connect them in the same order.
5. Label the image with prime marks.
6. Check: every vertex should move by the same arrow.

Worked Examples

Example 1: Translate a point

Translate P(2, -3) using the rule (x, y) \to (x - 6, y + 4).

1. Read the rule: a = -6 and b = 4. That is 6 left and 4 up.
2. Subtract 6 from the x-coordinate (see adding and subtracting integers): 2 - 6 = -4
3. Add 4 to the y-coordinate: -3 + 4 = 1
4. Write the image point: P'(-4, 1)

Answer: P'(-4, 1)

Example 2: Translate a triangle

Triangle ABC has vertices A(-4, 1), B(-1, 1) and C(-3, 4). Translate it 5 units right and 3 units down. This is the triangle in the graph above.

Write the rule first. Right 5 means add 5 to x. Down 3 means subtract 3 from y. So the rule is (x, y) \to (x + 5, y - 3).

Plot A', B' and C' and connect them. Check one side. AB runs from x = −4 to x = −1, so AB = 3. A'B' runs from x = 1 to x = 4, so A'B' = 3. The length stayed the same.

Answer: A'(1, -2), B'(4, -2), C'(2, 1)

Example 3: Find the rule from a point and its image

Point Q(3, -2) is translated to Q'(-1, 5). Write the rule. Then use it to translate R(0, 6).

1. Find a. Subtract the x-coordinates, image minus pre-image: a = -1 - 3 = -4
2. Find b. Subtract the y-coordinates, image minus pre-image: b = 5 - (-2)
3. Change subtracting a negative to adding: b = 5 + 2 = 7
4. Write the rule: (x, y) \to (x - 4, y + 7). That is 4 left and 7 up.
5. Translate R. Subtract 4 from the x-coordinate: 0 - 4 = -4
6. Add 7 to the y-coordinate: 6 + 7 = 13

Answer: The rule is (x, y) \to (x - 4, y + 7), and R'(-4, 13).

Example 4: Work backward to the pre-image

After the translation (x, y) \to (x + 3, y - 4), the image is D'(2, -5). Find the pre-image D.

To undo a translation, do the opposite move. The rule added 3 to x, so subtract 3. The rule subtracted 4 from y, so add 4.

1. Undo the x-move. Subtract 3: 2 - 3 = -1
2. Undo the y-move. Add 4: -5 + 4 = -1
3. Write the pre-image: D(-1, -1)
4. Check. Add 3 to x: -1 + 3 = 2
5. Check. Subtract 4 from y: -1 - 4 = -5. That gives D'(2, -5), so the answer works.

Answer: D(-1, -1)

Example 5: Word problem

A robot in a warehouse moves a shelf. On the floor plan, each grid unit is 1 meter. The shelf’s corners are at (1, 1), (3, 1), (3, 2) and (1, 2). The robot slides the shelf 6 meters east (right) and 8 meters north (up). Where are the new corners? How far does each corner travel?

The rule is (x, y) \to (x + 6, y + 8).

Every corner moves along the same arrow, so they all travel the same distance. Find the length of one arrow, from (1, 1) to (7, 9).

1. Use the distance formula on the move of 6 across and 8 up (see distance formula): d = \sqrt{6^2 + 8^2}
2. Square each number: d = \sqrt{36 + 64}
3. Add: d = \sqrt{100}
4. Take the square root: d = 10

Answer: The new corners are (7, 9), (9, 9), (9, 10) and (7, 10). Each corner travels 10 meters.

Common Mistakes

What a Translation Keeps the Same

Here are three more facts that help in later lessons.

• The arrows from each point to its image are all parallel and the same length. That is why one vector can describe the whole move.
• A translated line is parallel to the original line, or it is the same line. The slope does not change (see parallel and perpendicular lines).
• A translated angle has the same measure, so all the facts from measuring and classifying angles still hold.
• Graphs of functions can slide too. In Algebra 2, y = f(x - h) + k slides a parent function h units right and k units up.

Practice Problems

1. Translate A(5, 2) using (x, y) \to (x + 3, y - 7).
2. Translate B(-2, -6) using (x, y) \to (x - 4, y + 9).
3. Write a rule for a translation 6 units left and 2 units up.
4. Translate C(0, 4) by the vector \langle -1, -8 \rangle.
5. Triangle JKL has vertices J(1, 2), K(4, 2) and L(1, 6). Translate it using (x, y) \to (x - 5, y - 3).
6. Point M(-3, 4) is translated to M'(2, -1). Write the rule.
7. Use your rule from problem 6 to translate N(6, 0).
8. After (x, y) \to (x + 4, y - 6), the image is P'(-7, 2). Find the pre-image P.
9. In problem 5, find the lengths of JK and J'K'. What do you notice?
10. On a map, each grid unit is 1 kilometer. A ferry moves straight from (-2, -3) to (10, 2). Write the translation rule. Then find how far the ferry travels.

FAQ

What is a translation in geometry?

A translation is a slide. Every point of the figure moves the same distance in the same direction, with no turning or flipping.

What is the rule for a translation?

The rule is (x, y) \to (x + a, y + b). Add a to every x-coordinate and b to every y-coordinate.

Is a translation a rigid motion?

Yes. A translation keeps all lengths and angle measures, so the image is congruent to the pre-image.

How do you find the translation vector from two points?

Subtract the pre-image from the image. For Q(3, -2) and Q'(-1, 5), the vector is \langle -4, 7 \rangle.

Summary

• A translation slides every point the same distance in the same direction.
• The rule is (x, y) \to (x + a, y + b), or the vector \langle a, b \rangle.
• Right and up add. Left and down subtract.
• To find the rule, subtract: image minus pre-image.
• To undo a translation, use the opposite move.
• A translation is a rigid motion, so the image is congruent to the pre-image.

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