A Dilation Is An Isometry True False

A Dilation is an Isometry: True or False? A Deep Dive into Transformations

The statement "A dilation is an isometry" is false. Understanding why requires a thorough exploration of both dilations and isometries, their properties, and how they fundamentally differ. This article will delve into the mathematical definitions of these transformations, provide illustrative examples, and clarify the key distinctions that separate them.

Understanding Isometries: Preserving Distance and Shape

In geometry, an isometry is a transformation that preserves distances. This means that the distance between any two points in the original figure remains the same after the transformation. Consequently, isometries also preserve the shape and size of the figure. Common examples of isometries include:

• Translations: Shifting a figure along a vector without changing its orientation or size.
• Rotations: Turning a figure around a point, maintaining its shape and size.
• Reflections: Mirroring a figure across a line, preserving its shape and size.
• Glide Reflections: A combination of a reflection and a translation.

The critical characteristic of an isometry is the preservation of distance. This is a defining feature that sets it apart from other types of transformations, such as dilations.

Isometry Properties: A Summary

Let's summarize the key properties of isometries:

• Distance Preservation: The distance between any two points remains unchanged after the transformation.
• Shape Preservation: The shape of the figure remains identical.
• Size Preservation: The size (area or volume) of the figure remains unchanged.
• Angle Preservation: The angles within the figure remain unchanged.
• Parallelism Preservation: Parallel lines remain parallel after the transformation.

Understanding Dilations: Scaling and Proportionality

A dilation, on the other hand, is a transformation that changes the size of a figure but preserves its shape. It is defined by a center of dilation and a scale factor. The scale factor determines the amount of enlargement or reduction.

• Scale Factor > 1: The figure is enlarged.
• Scale Factor = 1: The figure remains unchanged (the identity transformation).
• 0 < Scale Factor < 1: The figure is reduced.
• Scale Factor < 0: The figure is enlarged or reduced and reflected across the center of dilation.

A dilation transforms a point P to a point P' such that the ratio of the distance from the center of dilation to P' and the distance from the center of dilation to P is equal to the scale factor.

Dilation Properties: A Contrasting View

Unlike isometries, dilations do not preserve distances. They alter the size of the figure, making them fundamentally different from isometries. Let's list the properties of dilations:

• Distance Non-Preservation: Distances between points are scaled by the scale factor.
• Shape Preservation: The shape of the figure remains similar to the original.
• Size Non-Preservation: The size (area or volume) of the figure is changed by the square (or cube) of the scale factor.
• Angle Preservation: Angles within the figure remain unchanged.
• Parallelism Preservation: Parallel lines remain parallel after the transformation.

The Crucial Difference: Distance Preservation

The key distinction between dilations and isometries lies in their effect on distances. Isometries preserve distances, while dilations do not. This difference is paramount in classifying these transformations. A dilation can change the length of every segment in a figure by the same scale factor, directly violating the distance preservation criterion of an isometry.

Consider a simple example: a square with side length 2 units. If we apply a dilation with a scale factor of 2, the resulting square will have side lengths of 4 units. The distance between opposite corners of the original square is 2√2 units, while the distance between the corresponding corners in the dilated square is 4√2 units. This clear change in distance demonstrates that the dilation is not an isometry.

Visualizing the Difference: Examples

Let's illustrate the difference with concrete examples.

Example 1: Isometry (Rotation)

Imagine a triangle with vertices at (1,1), (3,1), and (2,3). If we rotate this triangle 90 degrees counter-clockwise around the origin, the new vertices will be at (-1,1), (-1,3), and (-3,2). The distances between all vertices remain the same. This is a classic example of an isometry.

Example 2: Dilation

Now, let's take the same triangle and apply a dilation with a scale factor of 2 and a center of dilation at the origin. The new vertices will be at (2,2), (6,2), and (4,6). The distances between the vertices have all doubled. The distance between (1,1) and (3,1) is 2; after dilation, the distance between (2,2) and (6,2) is 4. This shows that the dilation is not an isometry.

Mathematical Proof of Non-Isometry

We can further solidify the concept with a mathematical proof. Let's consider two points, A and B, with coordinates (x₁, y₁) and (x₂, y₂) respectively. The distance between A and B is given by the distance formula:

d(A, B) = √((x₂ - x₁)² + (y₂ - y₁)²)

Under a dilation with center (0,0) and scale factor k, the points A and B are transformed to A' and B' with coordinates (kx₁, ky₁) and (kx₂, ky₂) respectively. The distance between A' and B' is:

d(A', B') = √((kx₂ - kx₁)² + (ky₂ - ky₁)²) = √(k²((x₂ - x₁)² + (y₂ - y₁)²)) = |k|√((x₂ - x₁)² + (y₂ - y₁)²) = |k|d(A, B)

Since the distance between A' and B' is |k| times the distance between A and B, unless k = 1 or k = -1 (which represent an identity transformation or a reflection-dilation respectively), the distances are not preserved. Therefore, a dilation (excluding trivial cases) is not an isometry.

Conclusion: Understanding the Distinctions

In summary, a dilation is not an isometry. While dilations preserve shape and angles, they fundamentally alter distances between points, a defining characteristic of isometries. Understanding the subtle but crucial difference between these two types of transformations is vital for a solid grasp of geometric principles and their applications in various fields. The mathematical proofs and illustrative examples presented clearly demonstrate why a dilation fails to satisfy the condition of distance preservation, thereby disqualifying it as an isometry. This distinction is critical in higher-level mathematics and related disciplines.