﻿ Definition of isometry in math with example diagrams and non-examples

# What is an Isometry?

#### What is an isometry?

Answer:
An isometry is a transformation that preserves distance.

Transformations that are isometries :

Type of transformation that is not an isometry :

Isometries can be classified as either direct or opposite, but more on that later.

### Diagrams of Isometries

Diagram 1

In the diagram below, both the image and the preimage of $$\triangle ABC$$ have the same dimensions, showing that reflections are isometries.

Diagram 2

Again in this diagram, both the image and the preimage of $$\triangle ABC$$ have the same dimensions, showing that translations are isometries.

Diagram 3

Lastly, both the image and the preimage of $$\triangle ABC$$ have the same dimensions, showing that rotations are isometries.

#### What is a direct isometry?

Answer:
A direct isometry preserves distance and preserves orientation.

Transformations that are direct isometries :

The type of transformations below are not direct isometries :

### Direct Isometry Diagrams

The animations below show that rotations and translations are direct isometries because the orientation as well as the distance is preserved.

Diagram 4
###### Rotation Example

In the animation below, the vetices of $$\triangle ABC$$ are in a clockwise order or orientation, and the same is true for the image after the rotation.

Diagram 5
###### Translation Example

Just like the pior animation, the vetices of $$\triangle ABC$$ are in a clockwise order or orientation, and the same is true for the image after the translation.

#### What is an opposite isometry?

Answer:

An opposite isometry preserves distance but changes the order, or orientation, from clockwise to counterclockwise, or vice versa.

The one type of transformation that is an opposite isometry is a reflection.

### Opposite Isometry Diagram

Diagram 6
###### Reflection Example

The animation below shows that reflections are opposite isometries because the orientation is reversed while the distance is preserved.