What Are Isometric Transformations?

In mathematics, a transformation is a rule that moves or changes a shape in some way. But not all transformations are created equal — some change the size or shape of a figure, while others keep it exactly the same.

Isometric Transformation

A transformation that preserves the size and shape of a figure. The image (result) is congruent to the original figure. The word isometric comes from the Greek words iso (equal) and metron (measure).

Think of it this way: if you could cut out a shape from paper and then move it, flip it, or spin it, the piece of paper hasn't changed — it's still the same size and shape. That's exactly what isometric transformations do.

There are three types of isometric transformations:

Translation (sliding)
Reflection (flipping)
Rotation (turning)

Warning

Dilation (enlargement or reduction) is NOT an isometric transformation because it changes the size of the figure. If a transformation changes distances between points, it is not isometric.

Here's a quick overview of the key differences — we'll explore each one in detail in the sections below:

Property	Translation	Reflection	Rotation
Shape & size preserved?	✓ Yes	✓ Yes	✓ Yes
Orientation preserved?	✓ Yes	✗ No (reversed)	✓ Yes
Described by	Vector	Mirror line	Centre, angle, direction

This table is your advance organiser — keep it in mind as you work through each transformation type.

Translation — Sliding a Shape

Translation

A transformation that moves every point of a figure the same distance in the same direction. The figure slides without rotating or flipping.

A translation is described using a translation vector, which tells you how far to move horizontally and vertically. We write it as:

$v = (b a)$

where $a$ is the horizontal shift (positive = right, negative = left) and $b$ is the vertical shift (positive = up, negative = down).

Analogy

Imagine sliding a book across a table. The book doesn't turn or flip — it just glides to a new position. Every corner of the book moves the same distance in the same direction. That's a translation!

Key properties of translations:

Every point moves the same distance and direction
Line segments stay the same length
Angles stay the same size
The figure does not rotate or flip
The image is congruent to the original

If a point $P (x, y)$ is translated by the vector $(b a)$ , its image is:
$P^{'} (x + a,; y + b)$

Example

Example: Translate triangle $A B C$ with vertices $A (1, 2)$ , $B (4, 2)$ , $C (1, 5)$ by the vector $(- 2 3)$ .

Step 1: Add 3 to each $x$ -coordinate and subtract 2 from each $y$ -coordinate.

$A^{'} = (1 + 3,; 2 - 2) = (4, 0)$
$B^{'} = (4 + 3,; 2 - 2) = (7, 0)$
$C^{'} = (1 + 3,; 5 - 2) = (4, 3)$

Step 2: Plot the new vertices and connect them. Triangle $A^{'} B^{'} C^{'}$ is the image, and it is congruent to triangle $A B C$ .

Translation — Sliding a Shape

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