As a geometry expert with a deep understanding of transformations and congruence, I can provide a comprehensive answer to your question regarding whether a dilation is an isometry.

Step 1: Understanding Isometries
An isometry is a type of geometric transformation that preserves the distance between any two points in a figure. This means that the shape and size of the figure remain unchanged after the transformation. There are several types of isometries, including:


1. Rotations: A rotation involves turning a figure around a fixed point, known as the center of rotation, through a certain angle without altering its size or shape.


2. Translations: A translation moves every point of a figure the same distance in a given direction. It does not change the figure's shape or size.


3. Reflections: A reflection flips a figure over a line, known as the line of reflection. The figure's size and shape remain the same, but its orientation is reversed.

Step 2: Defining Dilation
A dilation, also known as a scaling or expansion, is a transformation that changes the size of a figure but not its shape. It is achieved by multiplying the distance of each point in the figure from a fixed point, known as the center of dilation, by a scale factor. If the scale factor is greater than 1, the figure is enlarged. If it is less than 1, the figure is reduced in size.

**Step 3: Comparing Isometries and Dilation**
Now, let's compare the properties of isometries with those of a dilation to determine if a dilation can be considered an isometry:

- Size Preservation: Isometries maintain the size of the figure, whereas dilations alter it. This is a fundamental difference that disqualifies dilations as isometries.

- Shape Preservation: Both isometries and dilations preserve the shape of the figure. However, the key distinction lies in the fact that dilations change the size.

- Congruence: Isometries result in a congruent figure, meaning the original and the transformed figure are identical in both size and shape. Dilation, while preserving shape, does not preserve size, making the original and the transformed figure similar but not congruent.

Conclusion
Based on the analysis, a dilation is not an isometry because it does not preserve the size of the figure. While both transformations maintain the shape, the defining characteristic of an isometry is the preservation of both shape and size, which a dilation does not do.

Now, let's proceed to the next step as per your instructions.

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An isometry, such as a rotation, translation, or reflection, does not change the size or shape of the figure. A dilation is not an isometry since it either shrinks or enlarges a figure. ... An isometry is a transformation where the original shape and new image are congruent.read more >>