Applying the Dilation Rule

Oct 13, 2025 by windailys.com 48 views

Introduction

Dilation is a fundamental concept in geometry that involves changing the size of a figure while maintaining its shape. In this article, we will explore the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ and apply it to find the coordinates of vertex $V$ of the pre-image given the image STUV.

What is Dilation?

Dilation is a transformation that changes the size of a figure, but not its shape. It involves scaling the figure by a certain factor, which can be greater than, less than, or equal to 1. In the case of the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ , the figure is being scaled by a factor of $\frac{1}{3}$ .

The Dilation Rule

The dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ can be broken down into two parts:

The center of dilation is $O$ .
The scale factor is $\frac{1}{3}$ .

Applying the Dilation Rule

To apply the dilation rule, we need to multiply the coordinates of the pre-image by the scale factor. In this case, the scale factor is $\frac{1}{3}$ .

Finding the Coordinates of Vertex $V$

Given the image STUV, we need to find the coordinates of vertex $V$ of the pre-image. To do this, we can use the dilation rule and work backwards.

Let's assume that the coordinates of vertex $V$ of the pre-image are $(x, y)$ . Using the dilation rule, we can write:

\left(\frac{1}{3} x, \frac{1}{3} y\right) = V

To find the coordinates of vertex $V$ , we can multiply both coordinates by 3:

\left(3 \cdot \frac{1}{3} x, 3 \cdot \frac{1}{3} y\right) = (x, y)

Simplifying, we get:

(x, y) = (x, y)

This means that the coordinates of vertex $V$ of the pre-image are the same as the coordinates of vertex $V$ of the image.

Conclusion

In this article, we explored the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ and applied it to find the coordinates of vertex $V$ of the pre-image given the image STUV. We found that the coordinates of vertex $V$ of the pre-image are the same as the coordinates of vertex $V$ of the image.

Answer

The coordinates of vertex $V$ of the pre-image are:

A. (0, 0)

This is because the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ scales the figure by a factor of $\frac{1}{3}$ , which means that the coordinates of the pre-image are the same as the coordinates of the image.

Final Thoughts

Dilation is a fundamental concept in geometry that involves changing the size of a figure while maintaining its shape. In this article, we explored the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ and applied it to find the coordinates of vertex $V$ of the pre-image given the image STUV. We found that the coordinates of vertex $V$ of the pre-image are the same as the coordinates of vertex $V$ of the image.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Dilation" by Math Open Reference

Additional Resources

[1] Khan Academy: Dilation
[2] Mathway: Dilation

FAQs

Q: What is dilation?
A: Dilation is a transformation that changes the size of a figure, but not its shape.
Q: What is the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ ?
A: The dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ scales the figure by a factor of $\frac{1}{3}$ .
Q: How do I find the coordinates of vertex $V$ of the pre-image given the image STUV?
A: To find the coordinates of vertex $V$ of the pre-image, you can use the dilation rule and work backwards.
Dilation Q&A ================

Frequently Asked Questions

Q: What is dilation?

A: Dilation is a transformation that changes the size of a figure, but not its shape. It involves scaling the figure by a certain factor, which can be greater than, less than, or equal to 1.

Q: What is the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ ?

A: The dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ scales the figure by a factor of $\frac{1}{3}$ . This means that the coordinates of the pre-image are multiplied by $\frac{1}{3}$ to get the coordinates of the image.

Q: How do I find the coordinates of vertex $V$ of the pre-image given the image STUV?

A: To find the coordinates of vertex $V$ of the pre-image, you can use the dilation rule and work backwards. Multiply the coordinates of the image by the reciprocal of the scale factor to get the coordinates of the pre-image.

Q: What is the center of dilation?

A: The center of dilation is the point around which the dilation occurs. In the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ , the center of dilation is $O$ .

Q: What is the scale factor?

A: The scale factor is the factor by which the figure is scaled. In the dilation rule $D_{O, 1 / 3}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)$ , the scale factor is $\frac{1}{3}$ .

Q: How do I determine the scale factor?

A: To determine the scale factor, you can look at the coordinates of the image and the coordinates of the pre-image. The scale factor is the factor by which the coordinates of the pre-image are multiplied to get the coordinates of the image.

Q: What is the difference between dilation and translation?

A: Dilation changes the size of a figure, but not its shape. Translation changes the position of a figure, but not its size or shape.

Q: Can dilation be represented graphically?

A: Yes, dilation can be represented graphically. You can use a graph to show the dilation of a figure. The graph will show the original figure and the dilated figure.

Q: How do I apply dilation to a figure?

A: To apply dilation to a figure, you can use the dilation rule and multiply the coordinates of the figure by the scale factor.

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

A: Some real-world applications of dilation include:

Scaling up or down a design or blueprint
Creating a miniature or enlarged version of an object
Modeling the growth or shrinkage of an object

Q: How do I determine the coordinates of a dilated figure?

A: To determine the coordinates of a dilated figure, you can use the dilation rule and multiply the coordinates of the original figure by the scale factor.

Q: Can dilation be used to create a mirror image of a figure?

A: No, dilation cannot be used to create a mirror image of a figure. Dilation changes the size of a figure, but not its shape. To create a mirror image of a figure, you would need to use a different transformation, such as reflection.

Q: How do I determine the scale factor of a dilated figure?

A: To determine the scale factor of a dilated figure, you can look at the coordinates of the dilated figure and the coordinates of the original figure. The scale factor is the factor by which the coordinates of the original figure are multiplied to get the coordinates of the dilated figure.

Q: Can dilation be used to create a rotated figure?

A: No, dilation cannot be used to create a rotated figure. Dilation changes the size of a figure, but not its shape. To create a rotated figure, you would need to use a different transformation, such as rotation.

Q: How do I apply dilation to a 3D figure?

A: To apply dilation to a 3D figure, you can use the dilation rule and multiply the coordinates of the 3D figure by the scale factor.

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

A: Some common mistakes to avoid when applying dilation include:

Failing to multiply the coordinates of the figure by the scale factor
Using the wrong scale factor
Failing to account for the center of dilation

Q: How do I check my work when applying dilation?

A: To check your work when applying dilation, you can:

Verify that the coordinates of the dilated figure are correct
Check that the scale factor is correct
Make sure that the center of dilation is correct

Q: Can dilation be used to create a symmetrical figure?

A: Yes, dilation can be used to create a symmetrical figure. By applying dilation to a figure, you can create a symmetrical figure.

Q: How do I determine the coordinates of a dilated figure with a non-zero center of dilation?

A: To determine the coordinates of a dilated figure with a non-zero center of dilation, you can use the dilation rule and multiply the coordinates of the figure by the scale factor, then add the coordinates of the center of dilation.

Q: Can dilation be used to create a figure with a specific shape?

A: Yes, dilation can be used to create a figure with a specific shape. By applying dilation to a figure, you can create a figure with a specific shape.

Q: How do I apply dilation to a figure with a non-zero center of dilation?

A: To apply dilation to a figure with a non-zero center of dilation, you can use the dilation rule and multiply the coordinates of the figure by the scale factor, then add the coordinates of the center of dilation.

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

A: Some real-world applications of dilation in art include:

Creating a miniature or enlarged version of a painting or sculpture
Scaling up or down a design or blueprint
Modeling the growth or shrinkage of an object

Q: How do I determine the scale factor of a dilated figure with a non-zero center of dilation?

A: To determine the scale factor of a dilated figure with a non-zero center of dilation, you can look at the coordinates of the dilated figure and the coordinates of the original figure. The scale factor is the factor by which the coordinates of the original figure are multiplied to get the coordinates of the dilated figure.

Q: Can dilation be used to create a figure with a specific size?

A: Yes, dilation can be used to create a figure with a specific size. By applying dilation to a figure, you can create a figure with a specific size.

Q: How do I apply dilation to a figure with a specific size?

A: To apply dilation to a figure with a specific size, you can use the dilation rule and multiply the coordinates of the figure by the scale factor.

Q: What are some common mistakes to avoid when applying dilation in art?

A: Some common mistakes to avoid when applying dilation in art include:

Failing to multiply the coordinates of the figure by the scale factor
Using the wrong scale factor
Failing to account for the center of dilation

Q: How do I check my work when applying dilation in art?

A: To check your work when applying dilation in art, you can:

Verify that the coordinates of the dilated figure are correct
Check that the scale factor is correct
Make sure that the center of dilation is correct

Q: Can dilation be used to create a figure with a specific shape and size?

A: Yes, dilation can be used to create a figure with a specific shape and size. By applying dilation to a figure, you can create a figure with a specific shape and size.

Q: How do I apply dilation to a figure with a specific shape and size?

A: To apply dilation to a figure with a specific shape and size, you can use the dilation rule and multiply the coordinates of the figure by the scale factor.

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

A: Some real-world applications of dilation in architecture include:

Scaling up or down a design or blueprint
Creating a miniature or enlarged version of a building or structure
Modeling the growth or shrinkage of a building or structure

Q: How do I determine the scale factor of a dilated figure with a specific shape and size?

A: To determine the scale factor of a dilated figure with a specific shape and size, you can look at the coordinates of the dilated figure and the coordinates of the original figure. The scale factor is the factor by which the coordinates of the original figure are multiplied to get the coordinates of the dilated figure.

Q: Can dilation be used to create a figure with a specific orientation?

A: Yes, dilation can be used to create a figure with a specific orientation.