Which Of The Following Is A Dilation

Which of the Following is a Dilation? Understanding Transformations in Geometry

Understanding geometric transformations is crucial for anyone studying mathematics, particularly in geometry and related fields. This article delves into the concept of dilation, a specific type of transformation that changes the size of a geometric figure but preserves its shape. We'll explore what constitutes a dilation, differentiating it from other transformations like translation, rotation, and reflection, and provide a clear understanding of how to identify a dilation given different scenarios. By the end, you'll be able to confidently determine which transformations are dilations and which are not.

Introduction to Geometric Transformations

Geometric transformations involve manipulating geometric shapes, changing their position, size, or orientation on a coordinate plane. There are four primary types of transformations:

• Translation: A slide, moving a shape a certain distance horizontally and/or vertically without changing its size or orientation.
• Rotation: A turn, rotating a shape around a fixed point (center of rotation) by a specific angle.
• Reflection: A flip, creating a mirror image of the shape across a line of reflection.
• Dilation: A scaling, changing the size of a shape while maintaining its shape and proportionality.

Each transformation has specific properties and rules governing its application. Understanding these properties is key to distinguishing one transformation from another.

Understanding Dilation: The Scaling Transformation

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. It's all about scaling – enlarging or reducing the size of a figure. Key characteristics of a dilation include:

• Center of Dilation: This is a fixed point around which the dilation occurs. All points on the original figure are scaled proportionally relative to this center.
• Scale Factor: This is a numerical value (k) that determines the amount of scaling. A scale factor greater than 1 (k > 1) indicates an enlargement, while a scale factor between 0 and 1 (0 < k < 1) indicates a reduction. A scale factor of 1 (k = 1) results in no change (the image is congruent to the pre-image). A negative scale factor results in an enlargement or reduction and a reflection across the center of dilation.

The image produced by a dilation is called the dilation image or simply the image, while the original figure is called the pre-image. The relationship between corresponding points in the pre-image and image is directly proportional to the scale factor.

Identifying a Dilation: Practical Examples

Let's examine several examples to illustrate how to identify a dilation:

Example 1:

Consider two triangles, ΔABC and ΔA'B'C'. The coordinates of the vertices are:

A = (1, 1), B = (2, 3), C = (4, 1) A' = (2, 2), B' = (4, 6), C' = (8, 2)

To determine if this is a dilation, we need to check if the ratios of corresponding side lengths are constant and equal to the scale factor.

• AB = √((2-1)² + (3-1)²) = √5

• A'B' = √((4-2)² + (6-2)²) = √20 = 2√5

• BC = √((4-2)² + (1-3)²) = √8 = 2√2

• B'C' = √((8-4)² + (2-6)²) = √32 = 4√2

• CA = √((1-4)² + (1-1)²) = 3

• C'A' = √((2-8)² + (2-2)²) = 6

Notice that A'B'/AB = B'C'/BC = C'A'/CA = 2. This constant ratio indicates a dilation with a scale factor of 2. The center of dilation needs to be determined, which is beyond the scope of this basic example but involves solving simultaneous equations derived from the relationship between pre-image and image coordinates.

Example 2:

Consider a square with vertices (0,0), (2,0), (2,2), (0,2). A transformation results in a new square with vertices (0,0), (4,0), (4,4), (0,4). This is a dilation with a scale factor of 2 and center of dilation at (0,0).

Example 3 (Non-Dilation):

Consider a rectangle with vertices (1,1), (4,1), (4,3), (1,3). A transformation results in a rectangle with vertices (1,2), (4,5), (4,7), (1,6). This is not a dilation. While the shape is maintained, the sides are not scaled proportionally; the height has been scaled by a different factor than the width.

Example 4 (Non-Dilation):

A triangle is rotated 90 degrees clockwise around the origin. This is a rotation, not a dilation, as the size and shape of the triangle remain unchanged; only its orientation is altered.

Distinguishing Dilation from Other Transformations

It's crucial to distinguish dilation from other transformations. Here's a comparison:

A dilation is the only transformation that changes the size of a figure while preserving its shape and proportionality. Other transformations may change the orientation or position, but not the size and shape in a proportional manner.

The Mathematical Description of Dilation

Mathematically, a dilation with center (a, b) and scale factor k can be described using the following transformation rule:

(x, y) → (k(x - a) + a, k(y - b) + b)

This formula indicates how the coordinates of each point (x, y) in the pre-image are transformed into the corresponding coordinates in the image. If the center of dilation is the origin (0, 0), the formula simplifies to:

(x, y) → (kx, ky)

This simplified formula is commonly used when the center of dilation is at the origin.

Dilation in Different Geometrical Contexts

The concept of dilation is not limited to two-dimensional shapes. It can also be applied to three-dimensional figures and even higher dimensions. The principles remain the same: a proportional scaling of all dimensions relative to a center of dilation.

Frequently Asked Questions (FAQ)

Q: Can a dilation have a negative scale factor?

A: Yes, a negative scale factor indicates a dilation combined with a reflection across the center of dilation. The size changes proportionally, but the image is also flipped.

Q: What if the center of dilation is outside the figure?

A: The dilation still applies. Each point in the pre-image is scaled proportionally relative to the center of dilation, even if that center lies outside the figure.

Q: Can a dilation change the shape of a figure?

A: No. A true dilation preserves the shape of the figure. If the shape changes, it's not a dilation.

Q: What happens if the scale factor is 1?

A: If the scale factor is 1, the image is congruent to the pre-image (no change in size).

Q: How do I find the center of dilation?

A: Finding the center of dilation requires solving a system of equations using the coordinates of the pre-image and image points and the scale factor.

Conclusion

Understanding dilation is a fundamental aspect of geometric transformations. By recognizing the key characteristics – a change in size while maintaining shape and proportionality – you can confidently identify dilations and distinguish them from other transformations. Remember the critical role of the scale factor and the center of dilation. Mastering these concepts solidifies your understanding of geometric principles and lays a solid foundation for more advanced mathematical studies. The ability to identify dilations is important not only in pure mathematics, but also finds applications in fields like computer graphics, engineering, and architecture, where scaling and resizing are essential operations.