Understanding the Dilation of Triangle ABC

Oct 13, 2025 by windailys.com 127 views

Introduction

In geometry, a dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation that enlarges or reduces a figure by a scale factor. In this article, we will explore the effects of a dilation on triangle ABC, where A has the coordinates (-4, 3) and B has the coordinates (4, 4). We will examine the properties of the image triangle A'B'C' and determine what is true about it.

What is a Dilation?

A dilation is a transformation that maps a point (x, y) to a new point (kx, ky), where k is the scale factor. The scale factor determines the size of the image, with k > 1 indicating an enlargement and 0 < k < 1 indicating a reduction. In the case of triangle ABC, we are given the dilation $D_{0, 1/2}(x, y)$ , which means that the scale factor is 1/2.

Understanding the Dilation of Triangle ABC

To understand the dilation of triangle ABC, we need to apply the dilation transformation to each vertex of the triangle. Let's start with vertex A, which has the coordinates (-4, 3). Applying the dilation $D_{0, 1/2}(x, y)$ , we get:

A' = (1/2)(-4, 3) = (-2, 3/2)

Similarly, we can apply the dilation to vertex B, which has the coordinates (4, 4):

B' = (1/2)(4, 4) = (2, 2)

Finally, we can apply the dilation to vertex C, which has the coordinates (0, 0):

C' = (1/2)(0, 0) = (0, 0)

Properties of the Image Triangle A'B'C'

Now that we have the coordinates of the image vertices, we can examine the properties of the image triangle A'B'C'. Let's consider the following properties:

AB is parallel to A'B': Since the dilation is a similarity transformation, the image triangle A'B'C' is similar to the original triangle ABC. This means that the corresponding sides of the two triangles are proportional, and the corresponding angles are equal. Therefore, AB is parallel to A'B'.
A'B' is a dilation of AB: Since the dilation is a scale factor of 1/2, the length of A'B' is half the length of AB. This means that A'B' is a dilation of AB.
The image triangle A'B'C' is smaller than the original triangle ABC: Since the scale factor is 1/2, the image triangle A'B'C' is smaller than the original triangle ABC.
The image triangle A'B'C' has the same shape as the original triangle ABC: Since the dilation is a similarity transformation, the image triangle A'B'C' has the same shape as the original triangle ABC.

Conclusion

In conclusion, the image triangle A'B'C' has the following properties:

AB is parallel to A'B'.
A'B' is a dilation of AB.
The image triangle A'B'C' is smaller than the original triangle ABC.
The image triangle A'B'C' has the same shape as the original triangle ABC.

These properties are a result of the dilation transformation, which changes the size of the figure but not its shape. The image triangle A'B'C' is a smaller version of the original triangle ABC, but it retains the same shape and proportions.

Discussion

The dilation transformation is a fundamental concept in geometry, and it has many applications in various fields, such as art, architecture, and engineering. Understanding the properties of the image triangle A'B'C' can help us to better appreciate the effects of dilation on geometric figures.

In this article, we have explored the effects of a dilation on triangle ABC, where A has the coordinates (-4, 3) and B has the coordinates (4, 4). We have examined the properties of the image triangle A'B'C' and determined what is true about it. We have also discussed the importance of dilation in geometry and its applications in various fields.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Dilations and Similarity Transformations" by Math Open Reference
[3] "Geometry and Measurement" by National Council of Teachers of Mathematics

Q&A: A Dilation of Triangle ABC

Q: What is a dilation in geometry?

A: A dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation that enlarges or reduces a figure by a scale factor.

Q: What is the scale factor in the dilation $D_{0, 1/2}(x, y)$ ?

A: The scale factor in the dilation $D_{0, 1/2}(x, y)$ is 1/2. This means that the image triangle A'B'C' will be half the size of the original triangle ABC.

Q: What is the effect of the dilation on the coordinates of the vertices of triangle ABC?

A: The dilation $D_{0, 1/2}(x, y)$ changes the coordinates of the vertices of triangle ABC as follows:

A' = (1/2)(-4, 3) = (-2, 3/2)
B' = (1/2)(4, 4) = (2, 2)
C' = (1/2)(0, 0) = (0, 0)

Q: Is the image triangle A'B'C' similar to the original triangle ABC?

A: Yes, the image triangle A'B'C' is similar to the original triangle ABC. This means that the corresponding sides of the two triangles are proportional, and the corresponding angles are equal.

Q: Is AB parallel to A'B'?

A: Yes, AB is parallel to A'B'. This is a result of the dilation transformation, which preserves the parallelism of the sides of the triangle.

Q: Is the image triangle A'B'C' smaller than the original triangle ABC?

A: Yes, the image triangle A'B'C' is smaller than the original triangle ABC. This is a result of the scale factor of 1/2, which reduces the size of the triangle.

Q: Does the image triangle A'B'C' have the same shape as the original triangle ABC?

A: Yes, the image triangle A'B'C' has the same shape as the original triangle ABC. This is a result of the dilation transformation, which preserves the shape of the triangle.

Q: What are some real-world applications of dilation in geometry?

A: Dilation has many real-world applications in various fields, such as art, architecture, and engineering. For example, dilation can be used to create enlarged or reduced images of objects, or to design buildings and bridges with specific proportions.

Q: How can dilation be used in art and design?

A: Dilation can be used in art and design to create enlarged or reduced images of objects, or to design compositions with specific proportions. For example, an artist may use dilation to create a large or small version of a painting, or to design a composition with specific proportions.

Q: How can dilation be used in architecture and engineering?

A: Dilation can be used in architecture and engineering to design buildings and bridges with specific proportions. For example, an architect may use dilation to design a building with a specific ratio of height to width, or to design a bridge with a specific ratio of length to width.

Q: What are some common mistakes to avoid when working with dilation?

A: Some common mistakes to avoid when working with dilation include:

Failing to identify the scale factor
Failing to apply the dilation transformation correctly
Failing to check for similarity between the original and image triangles
Failing to consider the real-world applications of dilation

Q: How can dilation be used to solve problems in geometry?

A: Dilation can be used to solve problems in geometry by:

Creating enlarged or reduced images of objects
Designing compositions with specific proportions
Solving problems involving similarity and congruence
Designing buildings and bridges with specific proportions

Q: What are some tips for working with dilation?

A: Some tips for working with dilation include:

Always identify the scale factor
Always apply the dilation transformation correctly
Always check for similarity between the original and image triangles
Always consider the real-world applications of dilation

Conclusion

In conclusion, dilation is a fundamental concept in geometry that has many real-world applications. By understanding the properties of dilation and how to apply it correctly, you can solve problems in geometry and design compositions with specific proportions. Remember to always identify the scale factor, apply the dilation transformation correctly, and check for similarity between the original and image triangles.