As a field expert in geometry, I specialize in the study of shapes, sizes, positions, and transformations. One of the fundamental transformations in Euclidean geometry is an isometry, which preserves the distance between any two points. Let's delve into the concept of reflection and its relationship with isometry.
Reflection is a transformation that flips a shape over a line, known as the line of reflection. This line acts as a mirror, and the reflected image is a mirror image of the original shape across that line. Now, let's explore if a reflection is indeed an isometry and the reasons behind it.
Isometry is defined as a transformation that preserves the distance between any two points in a space. In simpler terms, it means that the shape and size of the figure remain unchanged after the transformation. There are several types of isometries, including translations, rotations, and reflections. Each of these transformations has specific properties that make them isometries.

1. Translations move every point of a figure the same distance in a given direction. It is an isometry because it does not change the shape or size of the figure.

2. Rotations spin a figure around a fixed point, known as the center of rotation, by a certain angle. It is also an isometry because it preserves the distances between points.

3. Reflections, as mentioned earlier, flip a figure over a line. The key property that makes a reflection an isometry is that it preserves the distance between points. When a figure is reflected, the distance between any two points in the original figure and their corresponding points in the reflected figure remains the same. This is because the line of reflection acts as an axis of symmetry. Every point and its reflection across this line are equidistant from the line.
Now, let's address the provided reference content. It mentions that translations, rotations, and reflections are examples of isometries. This is correct. It also introduces the concept of a glide reflection, which is a combination of a reflection followed by a translation in the direction of the reflection. This is also an isometry because it preserves distances between points.
However, the reference content also states that folding, cutting, or melting a sheet are not considered isometries. This is accurate because these actions do not preserve the distances between points. Folding can distort the shape and size of the figure, cutting can separate parts of the figure, and melting can change the figure's form entirely.
In conclusion, a reflection is an isometry because it preserves the distance between any two points in a figure. This property aligns with the definition of an isometry and distinguishes it from other transformations that do not preserve distances, such as folding, cutting, or melting.
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These are examples of translations, rotations, and reflections respectively. There is one further type of isometry, called a glide reflection (see below under classification of Euclidean plane isometries). However, folding, cutting, or melting the sheet are not considered isometries.read more >>