Finding Pre-Image Coordinates Given Dilation Rule A Step-by-Step Solution

Jul 12, 2025 by ADMIN 74 views

Understanding Dilation and Finding Pre-Image Coordinates

In the realm of geometry, dilation is a transformative process that alters the size of a figure without affecting its shape. This transformation is defined by a center of dilation and a scale factor. The center of dilation is a fixed point around which the figure is either enlarged or reduced, while the scale factor determines the extent of this enlargement or reduction. In this comprehensive guide, we will delve into the concept of dilation, exploring its properties and applications, and then apply this knowledge to solve a specific problem: determining the coordinates of a pre-image vertex given the dilation rule and the image coordinates.

Delving into the Concept of Dilation

Dilation is a fundamental concept in geometry that plays a crucial role in various fields, including computer graphics, cartography, and even art. Understanding dilation is essential for comprehending geometric transformations and their impact on shapes and sizes. At its core, dilation is a transformation that scales a figure proportionally with respect to a fixed point, known as the center of dilation. The scale factor, a numerical value, dictates the extent of this scaling. A scale factor greater than 1 indicates an enlargement, while a scale factor between 0 and 1 signifies a reduction. The center of dilation serves as the reference point for this scaling, with points farther from the center experiencing a greater shift than those closer to it.

The dilation rule provides a mathematical framework for understanding how dilation affects the coordinates of points. In the given problem, the dilation rule is expressed as $D_{O, 1 / 3}(x, y) ightarrowegin{pmatrix}rac{1}{3} x, \frac{1}{3} y

\end{pmatrix}$, where $D$ represents the dilation transformation, $O$ denotes the center of dilation (in this case, the origin), and $1/3$ is the scale factor. This rule essentially states that the coordinates of a point $(x, y)$ are transformed to $( rac{1}{3} x, rac{1}{3} y)$ after dilation. In simpler terms, both the x-coordinate and the y-coordinate are multiplied by the scale factor. This uniform scaling ensures that the shape of the figure remains unchanged, only its size is altered.

To grasp the concept of dilation more intuitively, consider a simple example. Imagine a triangle with vertices at (1, 1), (2, 1), and (1, 2). If we apply a dilation with a scale factor of 2 and a center of dilation at the origin, the vertices of the triangle would be transformed to (2, 2), (4, 2), and (2, 4), respectively. Notice that the triangle has doubled in size, but its overall shape remains the same. Conversely, if we apply a dilation with a scale factor of 1/2, the triangle would be reduced to half its original size. This fundamental principle of dilation – scaling the size while preserving the shape – is crucial in various applications.

Key Properties of Dilation

Dilation, as a geometric transformation, possesses several key properties that distinguish it from other transformations such as translations or rotations. These properties are essential for understanding how dilation affects geometric figures and for solving related problems. One of the most important properties of dilation is that it preserves the shape of the figure. This means that the angles and proportions within the figure remain unchanged after dilation. Only the size of the figure is altered, not its fundamental form. This shape-preserving property is what makes dilation a powerful tool in various applications, such as creating scaled versions of images or maps.

Another crucial property of dilation is that it maps lines to lines, rays to rays, and segments to segments. This means that if you dilate a line, the resulting image will also be a line. Similarly, a ray will be transformed into a ray, and a line segment will become another line segment. This property ensures that the basic geometric elements of a figure are preserved during dilation. However, it's important to note that the lengths of the segments will be scaled by the dilation factor. For instance, if a segment has a length of 5 units and you apply a dilation with a scale factor of 2, the resulting segment will have a length of 10 units.

The center of dilation plays a significant role in determining how dilation affects the position of points. Points that are farther from the center of dilation will experience a greater shift in position compared to points that are closer to the center. This is because the scaling effect is more pronounced for points that are located at a greater distance from the center. In the specific case where the center of dilation is the origin, the dilation transformation can be represented by a simple multiplication of the coordinates by the scale factor. This makes calculations relatively straightforward, as we will see in the problem-solving section.

Applications of Dilation

The applications of dilation extend far beyond the realm of pure geometry. This transformative process finds practical use in a variety of fields, impacting our daily lives in numerous ways. One prominent application of dilation is in computer graphics. When you zoom in or out on an image or a map on your computer or smartphone, you are essentially applying a dilation transformation. The image is scaled up or down proportionally, allowing you to see more detail or get a broader view. This functionality is crucial in various software applications, from image editing programs to mapping and navigation tools.

Cartography, the science of mapmaking, also relies heavily on dilation. Maps are scaled-down representations of the real world, and the process of creating a map involves dilation. Cartographers use scale factors to reduce the size of geographical features while preserving their shapes and proportions. This allows them to represent large areas on a manageable scale. Different maps may use different scale factors depending on their purpose. For example, a city map may have a larger scale factor than a world map, allowing for more detail to be shown.

Art and design are other areas where dilation plays a significant role. Artists and designers often use dilation to create scaled versions of their work or to manipulate the size and proportions of objects within a composition. For instance, an architect might use dilation to create blueprints of a building, scaling down the actual dimensions to fit on a piece of paper. Similarly, a graphic designer might use dilation to resize images or logos while maintaining their original proportions. The ability to scale objects proportionally is an essential tool in these creative fields.

Solving the Pre-Image Coordinate Problem

Now that we have a solid understanding of dilation, let's tackle the specific problem at hand: determining the coordinates of vertex V of the pre-image, given the dilation rule $D_{O, 1 / 3}(x, y) ightarrowegin{pmatrix}rac{1}{3} x, \frac{1}{3} y

\end{pmatrix}$ and the image S'T'U'V'. We are given that the coordinates of the image vertex V' are (0, 1/3). Our goal is to find the coordinates of the original vertex V before the dilation was applied.

The key to solving this problem lies in understanding that dilation is a reversible transformation. If we know the dilation rule and the coordinates of the image, we can reverse the process to find the coordinates of the pre-image. In this case, the dilation rule tells us that the coordinates of the image are obtained by multiplying the coordinates of the pre-image by the scale factor of 1/3. To reverse this process, we simply need to divide the coordinates of the image by the scale factor.

Mathematically, if the coordinates of V are (x, y), then the coordinates of V' after dilation are $( rac{1}{3} x, rac{1}{3} y)$ . We are given that V' is (0, 1/3), so we can set up the following equations:

\frac{1}{3} x = 0

\frac{1}{3} y = \frac{1}{3}

Solving the first equation for x, we multiply both sides by 3:

x = 0 * 3 = 0

Solving the second equation for y, we also multiply both sides by 3:

y = \frac{1}{3} * 3 = 1

Therefore, the coordinates of vertex V are (0, 1). This means that the original point V was located at (0, 1), and when dilated by a scale factor of 1/3 with the center of dilation at the origin, it was transformed to the point V' at (0, 1/3).

Step-by-Step Solution

To further clarify the solution process, let's break it down into a step-by-step guide:

Identify the dilation rule: The dilation rule is given as $D_{O, 1 / 3}(x, y) ightarrowegin{pmatrix}rac{1}{3} x, \frac{1}{3} y

\end{pmatrix}$. This tells us that the coordinates of the image are 1/3 of the coordinates of the pre-image. 2. Identify the coordinates of the image vertex: The coordinates of V' are given as (0, 1/3). 3. Set up equations: Let the coordinates of V be (x, y). Then, according to the dilation rule, we have the equations:

$
\frac{1}{3} x = 0
$
$
\frac{1}{3} y = \frac{1}{3}
$

Solve for x: Multiply both sides of the first equation by 3 to get:
$x = 0$
Solve for y: Multiply both sides of the second equation by 3 to get:
$y = 1$
Write the coordinates of the pre-image vertex: The coordinates of V are (0, 1).

Common Mistakes to Avoid

When solving dilation problems, there are a few common mistakes that students often make. Being aware of these potential pitfalls can help you avoid errors and arrive at the correct solution. One common mistake is failing to correctly interpret the dilation rule. It's crucial to understand that the dilation rule specifies how the coordinates of the pre-image are transformed to obtain the coordinates of the image. Confusing the direction of the transformation can lead to incorrect calculations.

Another common mistake is incorrectly applying the scale factor. Remember that the scale factor determines the extent of the scaling. If the scale factor is greater than 1, the figure is enlarged, and if it's between 0 and 1, the figure is reduced. It's essential to multiply or divide by the scale factor in the correct direction to find the pre-image or the image coordinates. In this specific problem, we needed to reverse the dilation, so we divided the image coordinates by the scale factor.

Finally, a lack of understanding of the properties of dilation can also lead to mistakes. Remember that dilation preserves the shape of the figure, but not the size. The center of dilation is a fixed point, and points farther from the center experience a greater shift in position. Keeping these properties in mind can help you visualize the transformation and avoid errors in your calculations.

Conclusion

In conclusion, understanding dilation is fundamental to grasping geometric transformations and their applications. Dilation, with its properties of shape preservation and proportional scaling, plays a vital role in various fields, from computer graphics to cartography. By understanding the dilation rule and its implications, we can effectively solve problems involving pre-image and image coordinates. In the specific problem we addressed, we successfully determined the coordinates of the pre-image vertex V by reversing the dilation transformation. By following a step-by-step approach and being mindful of common mistakes, we can confidently tackle dilation problems and deepen our understanding of geometric transformations.