Polygon ABCD Dilation To A'B'C'D' A Comprehensive Analysis

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Hey guys! Today, we're diving into a fascinating geometry problem involving polygon ABCD and how it transforms when we dilate it. We've got a polygon, specifically a rectangle, with its corners neatly defined on a coordinate plane. We'll explore how this rectangle changes its size when we apply a dilation, and most importantly, we'll figure out the scale factor that governs this transformation. So, let's get started and unlock the secrets of geometric dilation!

Understanding the Original Polygon ABCD

Our journey begins with understanding the original polygon ABCD. We are given the coordinates of its vertices: A(1, -1), B(3, -1), C(3, -2), and D(1, -2). Let's break down what this tells us. Each vertex is a point on a coordinate plane, defined by its x and y coordinates. Plotting these points, we can visualize the shape of our polygon. A is located at the point where x is 1 and y is -1. B sits at x = 3 and y = -1. C is at x = 3 and y = -2, and finally, D is found at x = 1 and y = -2. Connecting these points in the order A-B-C-D-A, we quickly realize that polygon ABCD is a rectangle. The sides AB and CD are horizontal lines (since the y-coordinates are the same), and the sides AD and BC are vertical lines (since the x-coordinates are the same). This rectangular shape is crucial for understanding how dilation will affect it.

Now, let's dig a little deeper into the dimensions of this rectangle. The length of side AB can be found by calculating the distance between points A and B. Since they lie on the same horizontal line, we simply subtract their x-coordinates: 3 - 1 = 2 units. Similarly, the length of side CD is also 2 units. The length of side AD (or BC) is found by subtracting the y-coordinates of A and D: -1 - (-2) = 1 unit. So, we have a rectangle with a width of 2 units and a height of 1 unit. This understanding of the original dimensions will be vital when we compare it to the dilated polygon. Visualizing this rectangle on a coordinate plane can be incredibly helpful. You can sketch it out on paper or use a graphing tool to get a clear picture of its shape and size. This visual representation will make it easier to grasp the concept of dilation and how it affects the polygon's dimensions.

The Dilated Polygon A'B'C'D'

Now, let's shift our focus to the dilated polygon A'B'C'D'. Dilation is a transformation that changes the size of a figure, either enlarging it or shrinking it. In our case, polygon ABCD has been dilated to create a new polygon, A'B'C'D', with new vertices. The coordinates of these new vertices are: A'(2, -2), B'(6, -2), C'(6, -4), and D'(2, -4). Just like we did with the original polygon, let's analyze these coordinates to understand the shape and dimensions of the dilated polygon. Point A' is located at x = 2 and y = -2. B' is at x = 6 and y = -2. C' is at x = 6 and y = -4, and D' is found at x = 2 and y = -4. Connecting these points, we can see that A'B'C'D' is also a rectangle. This is a key characteristic of dilation – it preserves the shape of the figure. A rectangle will always dilate into another rectangle, a square into a square, and so on. However, the size will change depending on the scale factor of the dilation.

Let's examine the dimensions of the dilated rectangle. The length of side A'B' is the distance between A' and B'. Since they are on the same horizontal line, we subtract their x-coordinates: 6 - 2 = 4 units. Similarly, the length of side C'D' is also 4 units. The length of side A'D' (or B'C') is found by subtracting the y-coordinates of A' and D': -2 - (-4) = 2 units. So, the dilated rectangle A'B'C'D' has a width of 4 units and a height of 2 units. Comparing these dimensions to the original rectangle, we can start to see how the dilation has affected the size. The original rectangle had a width of 2 units and a height of 1 unit, while the dilated rectangle has a width of 4 units and a height of 2 units. This suggests that the polygon has been enlarged, and we can now delve into calculating the scale factor to determine exactly how much larger it has become. The fact that the shape remains a rectangle is a crucial piece of information, confirming that we are dealing with a dilation and not some other type of transformation.

Determining the Scale Factor of Dilation

Now comes the exciting part: determining the scale factor of the dilation. The scale factor is the magic number that tells us how much the original figure has been enlarged or shrunk to create the dilated figure. It's the ratio of the corresponding side lengths of the dilated figure and the original figure. To find the scale factor, we can compare any pair of corresponding sides. Let's start by comparing the lengths of sides AB and A'B'. We already know that AB has a length of 2 units and A'B' has a length of 4 units. The scale factor is then the ratio of A'B' to AB, which is 4/2 = 2. This tells us that the polygon has been enlarged by a factor of 2.

To confirm this, let's compare another pair of corresponding sides, say AD and A'D'. AD has a length of 1 unit, and A'D' has a length of 2 units. The ratio of A'D' to AD is 2/1 = 2. Again, we get a scale factor of 2. This consistency is important – the scale factor should be the same no matter which pair of corresponding sides we compare. If we got different scale factors for different pairs of sides, it would indicate that the transformation is not a simple dilation. A scale factor of 2 means that every side of the original polygon has been multiplied by 2 to create the dilated polygon. This applies not only to the sides but also to the distances between any two points on the figure. The coordinates of the vertices have also been affected by this scale factor, as we'll see in the next section. Understanding the scale factor is key to understanding the entire dilation process. It allows us to predict how any point on the original figure will transform under the dilation.

Verifying the Scale Factor with Coordinates

To further solidify our understanding, let's verify the scale factor using the coordinates of the vertices. If the scale factor is indeed 2, then the coordinates of the vertices of A'B'C'D' should be twice the coordinates of the vertices of ABCD. However, there's a crucial detail we need to consider: the center of dilation. Dilation is always performed with respect to a center point. In this case, we're not explicitly given the center of dilation, but we can infer it. A simple way to check is to see if multiplying the coordinates of ABCD by 2 gives us the coordinates of A'B'C'D'. Let's try it:

  • 2 * A(1, -1) = (2, -2), which matches A'(2, -2)
  • 2 * B(3, -1) = (6, -2), which matches B'(6, -2)
  • 2 * C(3, -2) = (6, -4), which matches C'(6, -4)
  • 2 * D(1, -2) = (2, -4), which matches D'(2, -4)

It seems like multiplying the coordinates by 2 works perfectly! This strongly suggests that the center of dilation is the origin (0, 0). When the center of dilation is the origin, the coordinates of the dilated image are simply the original coordinates multiplied by the scale factor. This makes the calculation straightforward. However, it's important to remember that if the center of dilation were a different point, the calculation would be more complex. We would need to consider the distance and direction from the center of dilation to each point on the original figure. This example highlights the importance of understanding the center of dilation in the dilation process. While we were able to determine the scale factor by comparing side lengths, verifying it with the coordinates provides a more rigorous confirmation. This step-by-step approach ensures that we have a solid grasp of the transformation that has taken place.

Conclusion: The Dilation Transformation

Alright, guys, let's wrap things up! We've successfully navigated the world of dilation and analyzed the transformation of polygon ABCD into A'B'C'D'. We started by understanding the original polygon, a rectangle with vertices at A(1, -1), B(3, -1), C(3, -2), and D(1, -2). We then explored the dilated polygon, another rectangle with vertices at A'(2, -2), B'(6, -2), C'(6, -4), and D'(2, -4). The key to understanding this transformation was determining the scale factor, which we found to be 2. This means that the original rectangle has been enlarged by a factor of 2 to create the dilated rectangle. We confirmed this by comparing the side lengths of the two rectangles and also by verifying that the coordinates of the vertices of the dilated rectangle are twice the coordinates of the vertices of the original rectangle, indicating that the center of dilation is likely the origin.

This problem illustrates several important concepts about dilation. Firstly, dilation preserves the shape of the figure. A rectangle will always dilate into a rectangle, a triangle into a triangle, and so on. Secondly, the scale factor determines the amount of enlargement or reduction. A scale factor greater than 1 indicates an enlargement, while a scale factor between 0 and 1 indicates a reduction. Thirdly, the center of dilation is crucial for understanding the transformation. When the center of dilation is the origin, the calculations are simplified, but in general, the center of dilation plays a significant role in determining the position of the dilated image. This exploration of dilation provides a solid foundation for understanding more complex geometric transformations. By carefully analyzing the coordinates, side lengths, and scale factor, we can gain a deep understanding of how figures change under dilation. So, keep practicing, keep exploring, and you'll become a master of geometric transformations in no time!