Practice Maths

L61 — Translations

Key Terms

translation
A transformation that slides a shape a fixed distance in a fixed direction. Every point moves the same amount — no rotation or flipping.
image
The result of a transformation. Image points are labelled with a prime symbol: A becomes A′.
pre-image
The original shape before the transformation is applied.
column vector
A way to write a translation as (horizontal, vertical). Positive = right/up; negative = left/down.

What is a Translation?

A translation is a slide. Every point of the shape moves the same distance in the same direction. There is no rotation or flipping.

Translations are described as “right a, up b” (or left/down for negative values), or as a column vector.

The image (result) of a point A is written as A′ (read as “A prime”).

Properties Preserved by Translations

  • Same size and same shape.
  • Same orientation (not flipped or rotated).
  • Only the position changes.
A B C A′ B′ C′ right 4, down 1
Blue: original shape • Red dashed: translated image • Every vertex moves the same amount
Hot Tip: A translation NEVER changes size, shape, or orientation — only position. If the shape looks rotated or flipped, it is NOT a translation.

Worked Example 1 — Translate a Triangle

Translate the triangle with vertices A(1, 2), B(3, 2), C(2, 4) by right 3, down 2. Find the image vertices.

Rule: Add 3 to each x-coordinate; subtract 2 from each y-coordinate.

A(1, 2) → A′(1+3, 2−2) = A′(4, 0)

B(3, 2) → B′(3+3, 2−2) = B′(6, 0)

C(2, 4) → C′(2+3, 4−2) = C′(5, 2)

Worked Example 2 — Describe a Translation

Point P is at (2, 5). Its image P′ is at (7, 3). Describe the translation.

Horizontal change: 7 − 2 = +5 (right 5).

Vertical change: 3 − 5 = −2 (down 2).

Translation: right 5, down 2.

A B C A′ B′ C′ right 4, down 1

Blue = original shape. Red dashed = translated image. Each vertex moves the same amount.

What Is a Translation?

A translation is a transformation that slides a shape from one position to another, without rotating it, reflecting it, or changing its size. Every point in the shape moves the same distance in the same direction. The shape looks identical after a translation — it's just in a different position.

Think of sliding a book across a table: the book moves, but it doesn't spin or flip. That's a translation.

Describing a Translation

A translation is described by how far the shape moves in two directions: horizontally (left or right) and vertically (up or down).

  • "3 units right and 2 units up" — this is a translation description.
  • Using column vector notation: a translation of 3 right and 2 up is written as (3, 2) where the first number is the horizontal movement and the second is vertical.
  • Right and up are positive; left and down are negative.

For example, the vector (−4, 1) means "4 units left and 1 unit up."

Translating a Shape on a Coordinate Plane

To translate a shape on a grid, apply the same movement to every vertex (corner) of the shape:

  • Original point A = (2, 3). Translate by (4, −2). New point A′ = (2+4, 3−2) = (6, 1)
  • Original point B = (0, 5). Translate by (4, −2). New point B′ = (0+4, 5−2) = (4, 3)

After translating all vertices, connect them in the same order to draw the image.

Image vs Pre-image

In transformation geometry, the pre-image is the original shape, and the image is the result after the transformation. Image points are usually labelled with a prime symbol: A becomes A′, B becomes B′, etc. The image and pre-image are congruent (identical in shape and size) — only the position changes.

Key tip: In a translation, the size, shape, and orientation of the figure do NOT change — only the position changes. This means all sides keep the same length, all angles stay the same, and the shape doesn't flip or turn. If a transformation changes any of those things, it is NOT a translation.

  1. Translating Individual Points Fluency

    Find the new coordinates of each point after the given translation. Use the grid to check your thinking.

    1 2 3 4 −1 −2 −3 2 3 4 −1 −2 −3 P(1,2) P′(4,3) right 3, up 1
    Example: P(1,2) → P′(4,3)

    Find the image of each point:

    1. (3, 4) translated right 2, up 5.
    2. (1, 7) translated left 3, down 4.
    3. (0, 0) translated right 6, up 2.
    4. (−2, 3) translated right 5, down 1.

  2. Translating Shapes Fluency

    For each diagram, the original shape is drawn. Find and label the image vertices A′, B′, C′ (etc.) after the given translation. Check that the image has the same size and shape.

    2 4 6 −2 −4 2 4 6 −2 A B C A′ B′ C′
    (a) right 3, up 2
    2 4 −2 −4 2 −2 −4 P Q R S P′ Q′ R′ S′
    (b) left 2, down 3
    2 4 6 −2 2 4 −2 D E F D′ E′ F′
    (c) right 4, down 4
    2 4 6 −2 2 4 6 −2 J K L M J′ K′ L′ M′
    (d) left 3, up 1

  3. Describing Translations Fluency

    For each diagram, describe the translation that maps the blue point (original) to the red point (image). State how far right or left, and how far up or down.

    246 −2−4 246 −2−4 A(1, 2) A′(4, 5)
    (a) Describe the translation
    24 −2 2468 −2 B(3, 7) B′(1, 3)
    (b) Describe the translation
    2 −2−4 42 −2 C(0, 0) C′(−2, 4)
    (c) Describe the translation
    246 642 −2 D(5, 1) D′(5, 6)
    (d) Describe the translation

  4. Properties of Translations Understanding

    3 cm 5 cm 4 cm 3 cm ✓ 5 cm ✓ 4 cm ✓ translation Original Image ✓ same size ✓ same shape ✓ same orientation
    Translations preserve size, shape & orientation
    1. A triangle has side lengths 3 cm, 4 cm, and 5 cm. After a translation, what are the side lengths of the image?
    2. A shape is translated. Does its orientation change? Explain.
    3. True or False: After a translation, the image overlaps with the original shape.

    (d) Use the grid to help. Point A(2, 3) is translated to A′(5, 1). Point B(4, 6) is translated by the same translation. Find B′.

    246 36 14 A(2, 3) A′(5, 1) B(4, 6) B′ = ?
    A and A′ are shown — use the same translation to find B′

  5. Translation Puzzles Problem Solving

    2468 2468 1st 2nd 3rd ?
    (a) The pattern repeats: right 3, down 2. The first tile corner is at (0, 8). Where is the corner of the 4th tile?
    2468 246810 A′(5,6) B′(8,6) C′(6,9)
    (b) The red triangle A′B′C′ was made by translating the original right 4, up 3. Find original vertices A, B, C.
    24 46 0 P(1,4) +3,+1 P₁(4,5) −2,−5 P₂(?,?)
    (c) P(1,4) moves right 3, up 1, then left 2, down 5. Find the final coordinates and the single equivalent translation.
    246 37 Start(2,3) right 5 up 4 left 3 Final(?,?)
    (d) A ship starts at (2, 3) and travels: right 5, then up 4, then left 3. Find the final coordinates and total displacement from start.

  6. Column Vector Notation Fluency

    1234 −1 23 −1−2 2 1 ← right 2 ← up 1 P(1, 2) P′(3, 3) +2, +1
    Column vector: top = horizontal, bottom = vertical

    Translate each point using the given column vector:

    1. (2, 3)  — vector (4, 2): right 4, up 2
    2. (1, 5)  — vector (−3, 1): left 3, up 1
    3. (0, 0)  — vector (5, −4): right 5, down 4
    4. (−2, 4) — vector (2, −2): right 2, down 2

  7. Reverse Translation Understanding

    The image point (red) is shown. The translation that was applied is labelled. Find the original point by reversing the translation.

    246 42 A′(7,4) −3, −2 A = ?
    (a) Translation was right 3, up 2. Find A.
    24 621 B′(1,1) +4, +5 B = ?
    (b) Translation was left 4, down 5. Find B.
    −22 63 C′(0,6) −2, −3 C = ?
    (c) Translation was right 2, up 3. Find C.
    −2−4 2 −4 D′(−3,2) +1, −6 D = ?
    (d) Translation was left 1, up 6. Find D.

  8. Translations on a Coordinate Grid Understanding

    246 24 A B AB=4 A′ B′ A′B′=?
    (a) A(0,0) and B(4,0) are translated right 3, up 2. Find A′, B′ and length A′B′. Compare to AB = 4 units.
    2 51 M(2, 5) M′(2, 1) horizontal or vertical?
    (b) M(2, 5) is translated to M′(2, 1). Is this horizontal or vertical? Describe the translation fully.
    246 24 K L M K′ L′ M′
    (c) Describe the translation from the blue triangle to the red triangle. Verify it is the same for all three vertices.
    24 245 orig. (3, 5) ✓ image?
    (d) Square with vertices (0,0),(2,0),(2,2),(0,2). After a translation, one vertex maps to (3,5). Find the translation and all image vertices.

  9. Combined and Equivalent Translations Understanding

    start step 1 step 2 final equivalent single translation
    Two translations combine into one equivalent translation
    1. A shape is translated right 3, up 4, then right 2, down 1. Write the single equivalent translation.
    2. A shape is translated left 6, up 2, then right 4, up 3. Write the single equivalent translation.
    3. A shape is translated right 5, up 5, and then a second translation returns it to the start. Describe the second translation.

    (d) Use the grid to track each step. Point P(2, 3) undergoes three translations: right 4, up 1; then left 7, down 3; then right 1, up 2. What are the final coordinates? What single equivalent translation maps P to the final position?

    246 −2 43 1 P(2,3) +4,+1 −7,−3 +1,+2 P′(?,?)
    Follow the path of P through three translations to find the final position

  10. Translations in Context Problem Solving

    246 356 ♞(2,3) +1,+2 +2,+1 (?,?)
    (a) A chess knight starts at (2, 3) and makes two L-shaped jumps. What are its final coordinates?
    246810 1 1st 2nd 3rd?
    (b) The pattern repeats by translating right 4. The 1st and 2nd triangles are shown. Write the vertices of the 3rd triangle.

    (c) Farmer’s field

    A field has corners at (10, 5), (20, 5), (22, 12), (8, 12) in metres. The boundary is shifted 3 m right and 2 m up. Find the new corner positions.

    24 32−2 Start R4 U3 L2 D5 R1 U4 (?,?)
    (d) A robot follows the path shown. Find its final position and total distance travelled.
See Answers ➔