Geometry and Measurement · Position and Transformation · Lesson 3

Translating Points and Shapes by a Vector

When a trotro drives straight down the road, when you slide down a playground slide or push a drawer closed, everything moves along together without turning. That is a translation. In this lesson you will describe translations with column vectors, draw images on a coordinate plane and find the vector that moved a shape.

NaCCA B7.3.3.1.3 ⏱ about 45 min

By the end you will be able to

✔Explain a translation as a slide in which every point moves the same distance in the same direction.
✔Read and write a translation as a column vector (x-move over y-move).
✔Draw the image of a point or shape under a given translation vector and write the image coordinates.
✔Find the vector of a translation from an object point and its image, and describe the move in words.
✔State the properties of an image under translation: congruent, same orientation, sides parallel to the object's.

Key words

Translation: A transformation that slides every point of a shape the same distance in the same direction. The shape is not turned, flipped or resized.
Vector: A quantity with size and direction. A translation vector is written as a column: the top number is the move right (+) or left (−); the bottom number is the move up (+) or down (−).
Image: The shape after the transformation. Its corners are named with dashes: A′, B′, C′.
Congruent: Same shape and size. The image of a translation is congruent to the object.

Curriculum reference

Content standard: B7.3.3.1 Perform a single transformation (i.e. reflection and translation) on a 2D shape using graph paper (including technology) and describe the properties of the image under the transformation (i.e. congruence, similarity, etc.)

Indicator: Plot points and shapes (i.e. plane figures) on a coordinate plane and draw their images under translation by a given vector.

Starter

Down the slide at Independence Square

At the children's park, Kofi climbs to the top of the slide, sits and slides to the bottom. His head, hands and feet all moved the same distance in the same direction, and he is still facing forward. A mathematician says: Kofi has undergone a translation. If you stand on a map, you could say the slide moved him '4 steps east and 3 steps south'. The numbers 4 and 3 are the heart of this lesson.

1. What is a translation?

A translation is a slide. Every point of the shape moves the same distance in the same direction. Nothing is turned, flipped or made bigger or smaller, so the image looks exactly like the object, just in a new place. Moving a chair across the classroom without turning it, a lift going up a building, a trotro on a straight road and a drawer being pulled out are all translations.

▶ Every part of Kofi moves together. Notice that the red arrow shows both the direction and the distance.

We describe a translation with a column vector, written with two numbers one above the other. The top number is the horizontal move (right is positive, left is negative). The bottom number is the vertical move (up is positive, down is negative). For example $(3 - 2)$ means 3 units right and 2 units down, and $(- 4 5)$ means 4 units left and 5 units up.

Reading column vectors
Vector	Horizontal move	Vertical move	In words
$(50)$	5 right	none	5 units to the right
$(0 - 3)$	none	3 down	3 units down
$(- 2 4)$	2 left	4 up	2 left and 4 up
$(- 3 - 1)$	3 left	1 down	3 left and 1 down

2. Drawing the image of a point or shape

To translate a point (x, y) by the vector $(a b)$ , add the top number to x and the bottom number to y: the image is (x + a, y + b). For a shape, translate every corner by the same vector, plot the image corners and join them in the same order.

1Write down the coordinates of each corner of the object and the vector $(a b)$ .
2Add a to every x-coordinate and b to every y-coordinate.
3Plot the new points and join them in the same order, naming them A′, B′, C′ …
4Check: each image side must be parallel to and the same length as the object side.

▶ Translate the triangle by (5, −3): every corner moves 5 right and 3 down. The image is not turned.

▶ A square moved by (3, 4) goes right 3 and up 4. Notice the arrows are all parallel.

Properties of the image under translation: (1) it is congruent to the object (same lengths, angles and area); (2) it has the same orientation: the letters go round the same way, so it is not a mirror image; (3) each side of the image is parallel to the matching side of the object; (4) the lines joining each point to its image are parallel and equal in length.

3. Finding the vector and describing the translation

Sometimes you are given the object and the image and must describe the single movement. Pick one corner and its matching corner on the image, and subtract the object from the image: vector = (image x − object x, image y − object y). Count squares to check: right is positive, up is positive.

▶ The shape PART is translated to P′A′R′T′. Find P(1, 1) and P′(−2, −3): the move is 3 left and 4 down.

For the shape PART above, P is at (1, 1) and P′ is at (−2, −3). The vector is (−2 − 1, −3 − 1) = (−3, −4), that is, 3 units left and 4 units down. You must get the same vector from every pair of matching corners; use a second pair as your check. If the answers differ, the movement is not a translation, or you have matched the wrong corners.

How far did a point move? The distance is the length of the diagonal arrow. For $(34)$ the right-angled triangle has sides 3 and 4, so the distance is $3^{2} + 4^{2} = 5$ units (the 3-4-5 triangle). Likewise $(68)$ gives 10 units.

4. Undoing a translation and real-life uses

To undo a translation, move back the way you came: the reverse of $(a b)$ is $(- a - b)$ . If a shape moves by $(2 - 4)$ , then bringing it back needs $(- 2 4)$ . Doing both leaves the shape exactly where it started.

▶ Pattern makers repeat one motif by translating it. This shape moves 7 right and 2 up.

Real life: Pattern makers who print Adinkra cloth stamp the same motif again and again, sliding it a fixed distance each time. A carpet or tile design repeats one shape by translation. In computer games a character is moved across the screen by adding a vector to its coordinates 60 times a second. Surveyors use translation to copy the same plot shape to a new place on a map.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Translating a point

Find the image of A(2, 3) under translation by

(4 - 5)

Show the solution

Step 1Add the top number to x: 2 + 4 = 6.
Step 2Add the bottom number to y: 3 + (−5) = −2.
Step 3The image is (6, −2).

Answer: A′(6, −2)

Example 2

Translating a triangle

Triangle ABC has A(−3, 1), B(−1, 1) and C(−3, 4). Find A′B′C′ after translation by

(5 - 3)

▶ Each corner slides 5 right and 3 down.

Show the solution

Step 1Add 5 to each x-coordinate and −3 to each y-coordinate.
Step 2A(−3, 1) → (−3 + 5, 1 − 3) = (2, −2).
Step 3B(−1, 1) → (−1 + 5, 1 − 3) = (4, −2).
Step 4C(−3, 4) → (−3 + 5, 4 − 3) = (2, 1).
Step 5Check: AB = 2 and A′B′ = 4 − 2 = 2 ✓ (same length).

Answer: A′(2, −2), B′(4, −2), C′(2, 1)

Example 3

Describing the translation of PART

The shape PART has P(1, 1), A(3, 1), R(3, 3) and T(1, 2). Its image has P′(−2, −3). Describe the translation and find A′, R′ and T′. How far did each point move?

▶ Check the vector with a second pair of points: A(3, 1) → A′(0, −3) is also 3 left, 4 down.

Show the solution

Step 1Vector = P′ − P = (−2 − 1, −3 − 1) = (−3, −4). That is 3 units left and 4 units down.
Step 2A′ = (3 − 3, 1 − 4) = (0, −3).
Step 3R′ = (3 − 3, 3 − 4) = (0, −1).
Step 4T′ = (1 − 3, 2 − 4) = (−2, −2).
Step 5Distance moved = $3^{2} + 4^{2} = 25 = 5$ units.

Answer: Translation by (−3, −4); A′(0, −3), R′(0, −1), T′(−2, −2); each point moves 5 units

Example 4

Using one pair to move another point

A translation takes A(4, −2) to A′(1, 3). Find the vector and the image of B(2, −1).

Show the solution

Step 1Vector = (1 − 4, 3 − (−2)) = (−3, 5).
Step 2B′ = (2 + (−3), −1 + 5) = (−1, 4).
Step 3Check by counting on the grid: from A to A′ is 3 left and 5 up ✓.

Answer: Vector (−3, 5); B′(−1, 4)

Example 5

Working backwards

A translation by

(- 2 4)

moves a point to (5, −1). Find the original point.

▶ The marked point (7, −5) moves 2 left and 4 up to (5, −1).

Show the solution

Step 1The image is (x + (−2), y + 4) = (5, −1).
Step 2So x − 2 = 5, giving x = 7.
Step 3And y + 4 = −1, giving y = −5.
Step 4Or: apply the reverse vector $(2 - 4)$ to the image: (5 + 2, −1 − 4) = (7, −5) ✓.

Answer: (7, −5)

Summary

A translation slides every point the same distance in the same direction; no turning, flipping or resizing.
The vector $(a b)$ means a units right (left if negative) and b units up (down if negative).
The image of (x, y) is (x + a, y + b). Translate every corner and join in the same order.
To find the vector, subtract: image − object for x and y. Check with a second pair of points.
The image is congruent, same orientation and its sides are parallel to the object's sides. The reverse of (a, b) is (−a, −b).

Down the slide at Independence Square

1. What is a translation?

2. Drawing the image of a point or shape

3. Finding the vector and describing the translation

4. Undoing a translation and real-life uses

Worked examples

Translating a point

Translating a triangle

Describing the translation of PART

Using one pair to move another point

Working backwards

Summary

Practice: 15 questions