Isometric — Definition, Formula & Examples

Isometric means "same measure." A transformation is isometric when it moves a figure without changing its size or shape, so the original and the image are congruent.

An isometric transformation (or isometry) is a mapping of the plane that preserves distances between all pairs of points, meaning the pre-image and image are congruent figures.

How It Works

To check whether a transformation is isometric, compare the side lengths and angles of the original figure to those of the image. If every distance stays the same, the transformation is an isometry. Translations, reflections, rotations, and glide reflections are all isometric. Dilations and compressions change the size of a figure, so they are not isometric.

Worked Example

Problem: Triangle ABC has vertices A(1, 2), B(4, 2), and C(4, 6). It is reflected over the y-axis to produce triangle A'B'C'. Is this transformation isometric?

Step 1: Find the side lengths of the original triangle.

AB = 3,\quad BC = 4,\quad AC = \sqrt{(4-1)^2 + (6-2)^2} = 5

Step 2: Reflect each vertex over the y-axis by negating the x-coordinates: A'(-1, 2), B'(-4, 2), C'(-4, 6). Compute the new side lengths.

A'B' = 3,\quad B'C' = 4,\quad A'C' = 5

Step 3: Compare: every pair of corresponding side lengths is equal, so distances are preserved.

Answer: Yes, the reflection is isometric because all distances are preserved and the two triangles are congruent.

Why It Matters

Knowing which transformations are isometric helps you prove that two figures are congruent without measuring every side and angle. This idea appears throughout middle-school and high-school geometry, especially in congruence proofs and coordinate geometry problems.

Common Mistakes

Mistake: Thinking a dilation is isometric because the shape looks the same.

Correction: A dilation changes all distances by a scale factor, so it does not preserve size. Only translations, reflections, rotations, and glide reflections are isometric.