Finding Pre-Image Coordinates Under Dilation A Step-by-Step Guide

Jul 12, 2025 by ADMIN 66 views

Understanding Dilations and Finding Pre-Image Coordinates

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Dilations are transformations that resize figures, either enlarging or shrinking them, while maintaining their shape. In this article, we will explore the concept of dilation in detail, focusing on how to determine the coordinates of a pre-image vertex given the dilation rule and the image coordinates. We will specifically address the question: Given the dilation rule $D_{O, 1 / 3}(x, y) ightarrow\left(\frac{1}{3} x, \frac{1}{3} y\right)$ and the image S'T'U'V', what are the coordinates of vertex V of the pre-image? This problem requires us to understand the inverse relationship between dilation and pre-image coordinates, and to apply this knowledge to find the solution. Understanding dilations is crucial in various fields, including geometry, computer graphics, and image processing. Let's delve into the world of dilations and learn how to solve such problems effectively.

What is Dilation?

Dilation, in the realm of geometry, is a transformation that alters the size of a figure without affecting its shape. Imagine taking a photograph and zooming in or out – that’s essentially what dilation does to a geometric figure. The dilation is defined by two key components: the center of dilation and the scale factor. The center of dilation is a fixed point around which the figure is either enlarged or reduced. The scale factor, often denoted by 'k', determines the extent of the resizing. If 'k' is greater than 1, the figure is enlarged, and if 'k' is between 0 and 1, the figure is reduced. When k is negative, the figure is dilated and reflected through the center of dilation.

To put it simply, if we have a point (x, y) and we apply a dilation with a scale factor of 'k' centered at the origin (0, 0), the new coordinates (x', y') of the dilated point are given by (kx, ky). This means each coordinate of the original point is multiplied by the scale factor. The transformation rule for dilation centered at the origin is therefore:

$D_{O, k}(x, y) ightarrow (kx, ky)$

Understanding this basic principle is crucial for solving problems related to dilations, including finding the coordinates of pre-images, which we will explore in detail later.

The center of dilation plays a critical role in how the dilation affects the figure. If the center of dilation is within the figure, the figure will expand or contract outwards or inwards from that point. If the center is outside the figure, the dilation will cause the figure to move away from or towards the center. For instance, consider a triangle dilated from a center outside the triangle; the dilated image will appear to shift away from the center if the scale factor is greater than 1, and towards the center if the scale factor is between 0 and 1.

The scale factor dictates the magnitude of the change in size. A scale factor of 2, for example, doubles the size of the figure, while a scale factor of 0.5 reduces the size by half. A scale factor of 1 implies no change in size, as the image is congruent to the pre-image. Understanding the scale factor is vital for calculating the new dimensions and positions of the dilated figure. In real-world applications, dilations are used extensively in computer graphics to zoom in and out of images, in architecture to scale building plans, and in various other fields where resizing objects while maintaining their shape is necessary. Thus, a solid grasp of dilation principles is beneficial across numerous disciplines.

The Dilation Rule and Pre-Images

In the given problem, we are presented with the dilation rule $D_{O, 1 / 3}(x, y) ightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ . This notation tells us that the dilation is centered at the origin (O) and has a scale factor of $\frac{1}{3}$ . This means that every point (x, y) in the original figure (the pre-image) is transformed to a new point $\left(\frac{1}{3} x, \frac{1}{3} y\right)$ in the dilated image.

The dilation rule provides a clear mathematical relationship between the coordinates of the pre-image and the image. Specifically, it states that the x-coordinate of the image is $\frac{1}{3}$ times the x-coordinate of the pre-image, and similarly, the y-coordinate of the image is $\frac{1}{3}$ times the y-coordinate of the pre-image. Understanding this rule is fundamental to solving the problem at hand, which requires us to find the coordinates of a vertex in the pre-image.

The concept of a pre-image is crucial in understanding transformations like dilation. The pre-image is the original figure before the transformation is applied, while the image is the figure after the transformation. In our case, we are given the image S'T'U'V' and we need to find the pre-image STUV. The dilation rule helps us move from the pre-image to the image, but to find the pre-image coordinates, we need to reverse this process.

To find the pre-image coordinates, we essentially need to undo the dilation. Since the dilation rule multiplies the coordinates by the scale factor, the inverse operation would be to divide the coordinates of the image by the scale factor. However, dividing by a fraction is the same as multiplying by its reciprocal. In this case, the scale factor is $\frac{1}{3}$ , so its reciprocal is 3. Therefore, to find the coordinates of vertex V in the pre-image, we need to multiply the coordinates of vertex V' in the image by 3.

This inverse relationship is a key concept in transformations. Understanding how to reverse a transformation allows us to move between the image and pre-image, which is a valuable skill in geometry and related fields. For example, in computer graphics, this principle is used to map textures onto 3D models, where the pre-image represents the 2D texture and the image represents its appearance on the 3D surface. Similarly, in image processing, reversing transformations can help in restoring distorted images to their original form. Thus, the ability to work with both the forward and reverse transformations is essential for a comprehensive understanding of geometric transformations.

Finding the Coordinates of Vertex V

Now, let’s apply our understanding to the specific problem. We are given the dilation rule $D_{O, 1 / 3}(x, y) ightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ and we need to find the coordinates of vertex V of the pre-image, given that we have the image S'T'U'V'. To find the coordinates of V, we need to reverse the dilation process. As discussed earlier, this means multiplying the coordinates of V' by the reciprocal of the scale factor, which is 3.

From the problem statement, we are given the options for the coordinates of vertex V as:

A. (0, 0) B. $\left(0, \frac{1}{3}\right)$ C. (0, 1) D. (0, 3)

To determine the correct answer, we need to consider what the coordinates of V' would be after the dilation is applied. Let's assume V' has coordinates (x', y'). According to the dilation rule, the coordinates of V' are given by:

x' = $\frac{1}{3}$ x y' = $\frac{1}{3}$ y

Where (x, y) are the coordinates of V. To find (x, y), we reverse the equations:

x = 3x' y = 3y'

Now, let's analyze each option for the coordinates of V:

A. (0, 0): If V is (0, 0), then V' would be $\left(\frac{1}{3}(0), \frac{1}{3}(0)\right)$ = (0, 0). This is a possible solution.
B. $\left(0, \frac{1}{3}\right)$ : If V is $\left(0, \frac{1}{3}\right)$ , then V' would be $\left(\frac{1}{3}(0), \frac{1}{3}(\frac{1}{3})\right)$ = $\left(0, \frac{1}{9}\right)$ . To reverse this, we multiply the coordinates of V' by 3, which gives us (0, $\frac{1}{3}$ ), matching the option.
C. (0, 1): If V is (0, 1), then V' would be $\left(\frac{1}{3}(0), \frac{1}{3}(1)\right)$ = $\left(0, \frac{1}{3}\right)$ . To reverse this, we multiply the coordinates of V' by 3, which gives us (0, 1), matching the option.
D. (0, 3): If V is (0, 3), then V' would be $\left(\frac{1}{3}(0), \frac{1}{3}(3)\right)$ = (0, 1). To reverse this, we multiply the coordinates of V' by 3, which gives us (0, 3), matching the option.

Since we are looking for the pre-image coordinates, we need to find the coordinates of V by multiplying the coordinates of V' by 3. Let's denote the coordinates of V' as (0, y'). Then, according to the dilation rule, $\frac{1}{3}$ y = y', so y = 3y'. Now, we check the options:

If V' is at (0,1), then V is (0,3)

Therefore, the correct answer is D. (0, 3).

This step-by-step analysis demonstrates how to apply the inverse dilation rule to find the pre-image coordinates. By understanding the relationship between the dilation rule, the scale factor, and the coordinates of the image and pre-image, we can effectively solve such problems.

Conclusion

In conclusion, understanding the principles of dilation and the relationship between pre-images and images is crucial for solving geometric problems involving transformations. We explored the dilation rule $D_{O, 1 / 3}(x, y) ightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ and successfully determined the coordinates of vertex V of the pre-image by applying the inverse dilation operation. This involved multiplying the coordinates of the image vertex V' by the reciprocal of the scale factor.

The key takeaways from this discussion are:

Dilation is a transformation that resizes a figure using a center of dilation and a scale factor.
The dilation rule provides the mathematical relationship between the pre-image and image coordinates.
To find the pre-image coordinates, we reverse the dilation process by multiplying the image coordinates by the reciprocal of the scale factor.

By mastering these concepts, you can confidently tackle a wide range of problems involving dilations and other geometric transformations. The principles discussed here are not only fundamental to geometry but also have applications in various fields, including computer graphics, image processing, and engineering. Continued practice and application of these concepts will further enhance your understanding and problem-solving skills in mathematics and related disciplines.

Thus, the problem of finding the pre-image coordinates after a dilation highlights the importance of understanding inverse transformations. This concept extends beyond dilations and applies to other transformations like rotations, reflections, and translations. In essence, understanding how to reverse a transformation allows us to analyze and manipulate geometric figures effectively, making it a cornerstone of geometric problem-solving.