Does a translation change the size or orientation of a figure?

No. A translation is a rigid motion (isometry), so the image is congruent to the original. Side lengths, angle measures, and the order of vertices (orientation) all remain unchanged.

Translation (Geometry) — Definition, Formula & Examples

A translation is a geometry transformation that slides every point of a figure the same distance in the same direction, without rotating, reflecting, or resizing it. The shape and size of the figure stay exactly the same — only its position changes.

A translation is an isometric transformation defined by a vector $\vec{v} = \langle a, b \rangle$ that maps each point $P(x, y)$ in the plane to a unique image point $P'(x + a, y + b)$ . Because every point moves by the same vector, distances, angle measures, and orientation are all preserved.

Key Formula

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

Where:

$x, y$ = Coordinates of any point on the original figure (pre-image)
$a$ = Horizontal shift (positive = right, negative = left)
$b$ = Vertical shift (positive = up, negative = down)

How It Works

To translate a figure, you add the same horizontal shift and vertical shift to every point. These shifts are described by a translation vector

\langle a, b \rangle

, where

a

is the horizontal change and

b

is the vertical change. A positive

a

moves the figure right, while a negative

a

moves it left. A positive

b

moves it up, and a negative

b

moves it down. After translating, the new figure (called the image) is congruent to the original (called the pre-image).

Worked Example

Problem: Triangle ABC has vertices A(1, 2), B(4, 2), and B(4, 6). Translate the triangle by the vector ⟨−3, 5⟩. Find the vertices of the image.

Step 1: Apply the translation rule to point A by adding −3 to the x-coordinate and 5 to the y-coordinate.

A'(1 + (-3),\; 2 + 5) = A'(-2, 7)

Step 2: Apply the same rule to point B.

B'(4 + (-3),\; 2 + 5) = B'(1, 7)

Step 3: Apply the same rule to point C.

C'(4 + (-3),\; 6 + 5) = C'(1, 11)

Answer: The image triangle A'B'C' has vertices A'(−2, 7), B'(1, 7), and C'(1, 11).

Another Example

Problem: Point P(−5, 3) is translated to P'(2, −1). What translation vector was used?

Step 1: Find the horizontal shift by subtracting the original x from the image x.

a = 2 - (-5) = 7

Step 2: Find the vertical shift by subtracting the original y from the image y.

b = -1 - 3 = -4

Answer: The translation vector is ⟨7, −4⟩, meaning the point moved 7 units right and 4 units down.

Visualization

Why It Matters

Translations appear throughout middle-school and high-school geometry courses whenever you study congruence and coordinate transformations. In computer graphics and video-game design, every time a character or object moves across the screen without turning, that motion is a translation. Understanding translations also builds the foundation for combining transformations, such as glide reflections.

Common Mistakes

Mistake: Subtracting instead of adding the translation values (or mixing up the sign).

Correction: Always add the vector components: x + a and y + b. If the vector is ⟨−3, 5⟩, you add −3 (which effectively subtracts 3) to x and add 5 to y.

Mistake: Applying different shift amounts to different points of the figure.

Correction: Every single point must be shifted by the same vector. If even one point gets a different shift, the figure's shape will be distorted, and it is no longer a translation.