Translation (Geometry)

A translation is a transformation that moves every point of a shape the same distance in the same direction. The figure doesn't rotate, flip, or change size — it simply slides from one position to another.

A translation is a rigid transformation (isometry) that maps every point $P(x, y)$ of a figure to a new point $P'(x + a, y + b)$ , where $a$ and $b$ are constants representing the horizontal and vertical components of the translation vector. Because every point moves by the same vector, the shape and size of the figure are preserved, and corresponding sides remain parallel.

Key Formula

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

Where:

$(x, y)$ = the coordinates of the original point
$(x + a, y + b)$ = the coordinates of the translated point (the image)
$a$ = the horizontal shift (positive = right, negative = left)
$b$ = the vertical shift (positive = up, negative = down)

Worked Example

Problem: Triangle ABC has vertices A(1, 3), B(4, 3), and B(4, 7). Translate the triangle 5 units to the right and 2 units down. Find the coordinates of the image A'B'C'.

Step 1: Identify the translation vector. Moving 5 right means

a = 5

. Moving 2 down means

b = -2

. The rule is:

(x, y) \rightarrow (x + 5,\; y - 2)

Step 2: Apply the rule to point A(1, 3).

A' = (1 + 5,\; 3 - 2) = (6, 1)

Step 3: Apply the rule to point B(4, 3).

B' = (4 + 5,\; 3 - 2) = (9, 1)

Step 4: Apply the rule to point C(4, 7).

C' = (4 + 5,\; 7 - 2) = (9, 5)

Answer: The image triangle A'B'C' has vertices A'(6, 1), B'(9, 1), and C'(9, 5).

Visualization

Why It Matters

Translations show up whenever something moves without turning or resizing — scrolling text on a screen, shifting a design element in graphic software, or describing the motion of an object along a straight path in physics. In geometry, translations are one of the four basic rigid motions used to define congruence: two figures are congruent if one can be mapped onto the other through a sequence of translations, reflections, and rotations.

Common Mistakes

Mistake: Mixing up the sign of the vertical shift — adding when you should subtract, or vice versa.

Correction: "Down" and "left" are negative directions on the coordinate plane. A shift of 2 units down means

b = -2

, so you subtract 2 from the

y

-coordinate, not add.

Mistake: Applying the translation to only some vertices and forgetting the rest.

Correction: Every point in the figure must move by the same vector. Apply the rule

(x + a,\; y + b)

to all vertices to get the correct image.

Related Terms

Transformations — Translation is one type of transformation
Isometry — Translations preserve distance and are isometries
Reflection — Another rigid motion that flips a figure
Rotation — Another rigid motion that turns a figure