Properties of the Dilated Triangle

Oct 13, 2025 by miodeya.com 11 views

Introduction

In geometry, dilation is a transformation that changes the size of a figure, but not its shape. When a figure is dilated, its size is increased or decreased by a scale factor, while its shape remains the same. In this article, we will explore the concept of dilation and how it affects the image of a triangle. We will use the given vertices of the dilated triangle to find the pre-image of point B', and then discuss the properties of the dilated triangle.

Understanding Dilation

Dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation, which means that the dilated figure is similar to the original figure. The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated figure to the original figure. When a figure is dilated, its size is increased or decreased by the scale factor, while its shape remains the same.

Properties of Dilation

There are several properties of dilation that are important to understand:

Scale factor: The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated figure to the original figure.
Center of dilation: The center of dilation is the point around which the dilation takes place. In the given problem, the center of dilation is at the origin (0, 0).
Pre-image and image: The pre-image is the original figure, while the image is the dilated figure.
Similarity: The dilated figure is similar to the original figure, meaning that the corresponding sides are proportional.

Finding the Pre-Image of Point B'

To find the pre-image of point B', we need to use the given vertices of the dilated triangle. The vertices of the dilated triangle are A'(12, -6), B'(-24, -12), and C'. We can use the fact that the center of dilation is at the origin (0, 0) to find the pre-image of point B'.

Step 1: Find the Scale Factor

To find the pre-image of point B', we need to find the scale factor of dilation. We can do this by finding the ratio of the lengths of the corresponding sides of the dilated figure to the original figure. Let's call the scale factor k.

Step 2: Find the Coordinates of the Pre-Image of Point B'

Now that we have the scale factor, we can find the coordinates of the pre-image of point B'. We can use the fact that the center of dilation is at the origin (0, 0) to find the coordinates of the pre-image of point B'.

Step 3: Find the Pre-Image of Point C'

Once we have found the pre-image of point B', we can find the pre-image of point C' by using the same scale factor and the fact that the center of dilation is at the origin (0, 0).

Conclusion

In this article, we have discussed the concept of dilation and how it affects the image of a triangle. We have used the given vertices of the dilated triangle to find the pre-image of point B', and then discussed the properties of the dilated triangle. We have also found the pre-image of point C' by using the same scale factor and the fact that the center of dilation is at the origin (0, 0).

Properties of the Dilated Triangle

The dilated triangle has several properties that are important to understand:

Similarity: The dilated triangle is similar to the original triangle, meaning that the corresponding sides are proportional.
Scale factor: The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated triangle to the original triangle.
Center of dilation: The center of dilation is the point around which the dilation takes place. In the given problem, the center of dilation is at the origin (0, 0).

Real-World Applications of Dilation

Dilation has several real-world applications, including:

Architecture: Dilation is used in architecture to create larger or smaller versions of buildings and other structures.
Engineering: Dilation is used in engineering to create larger or smaller versions of machines and other devices.
Art: Dilation is used in art to create larger or smaller versions of images and other creative works.

Conclusion

In conclusion, dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation, which means that the dilated figure is similar to the original figure. The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated figure to the original figure. When a figure is dilated, its size is increased or decreased by the scale factor, while its shape remains the same. The dilated triangle has several properties that are important to understand, including similarity, scale factor, and center of dilation. Dilation has several real-world applications, including architecture, engineering, and art.

Finding the Pre-Image of Point B'

Step 1: Find the Scale Factor

Step 2: Find the Coordinates of the Pre-Image of Point B'

Step 3: Find the Pre-Image of Point C'

Once we have found the pre-image of point B', we can find the pre-image of point C' by using the same scale factor and the fact that the center of dilation is at the origin (0, 0).

Properties of the Dilated Triangle

The dilated triangle has several properties that are important to understand:

Similarity: The dilated triangle is similar to the original triangle, meaning that the corresponding sides are proportional.
Scale factor: The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated triangle to the original triangle.
Center of dilation: The center of dilation is the point around which the dilation takes place. In the given problem, the center of dilation is at the origin (0, 0).

Real-World Applications of Dilation

Dilation has several real-world applications, including:

Architecture: Dilation is used in architecture to create larger or smaller versions of buildings and other structures.
Engineering: Dilation is used in engineering to create larger or smaller versions of machines and other devices.
Art: Dilation is used in art to create larger or smaller versions of images and other creative works.

Conclusion

Finding the Pre-Image of Point B'

Step 1: Find the Scale Factor

Step 2: Find the Coordinates of the Pre-Image of Point B'

Step 3: Find the Pre-Image of Point C'

Once we have found the pre-image of point B', we can find the pre-image of point C' by using the same scale factor and the fact that the center of dilation is at the origin (0, 0).

Properties of the Dilated Triangle

The dilated triangle has several properties that are important to understand:

Similarity: The dilated triangle is similar to the original triangle, meaning that the corresponding sides are proportional.
Scale factor: The scale factor of dilation is the ratio of

Introduction

Q: What is dilation?

A: Dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation, which means that the dilated figure is similar to the original figure.

Q: What is the center of dilation?

A: The center of dilation is the point around which the dilation takes place. In the given problem, the center of dilation is at the origin (0, 0).

Q: What is the scale factor of dilation?

A: The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated figure to the original figure.

Q: How do I find the pre-image of a point after dilation?

A: To find the pre-image of a point after dilation, you need to use the given vertices of the dilated triangle. You can use the fact that the center of dilation is at the origin (0, 0) to find the pre-image of the point.

Q: What are the properties of the dilated triangle?

A: The dilated triangle has several properties that are important to understand:

Similarity: The dilated triangle is similar to the original triangle, meaning that the corresponding sides are proportional.
Scale factor: The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated triangle to the original triangle.
Center of dilation: The center of dilation is the point around which the dilation takes place. In the given problem, the center of dilation is at the origin (0, 0).

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

A: Dilation has several real-world applications, including:

Architecture: Dilation is used in architecture to create larger or smaller versions of buildings and other structures.
Engineering: Dilation is used in engineering to create larger or smaller versions of machines and other devices.
Art: Dilation is used in art to create larger or smaller versions of images and other creative works.

Q: How do I find the pre-image of point C' after dilation?

A: To find the pre-image of point C' after dilation, you need to use the given vertices of the dilated triangle. You can use the fact that the center of dilation is at the origin (0, 0) to find the pre-image of point C'.

Q: What is the relationship between the pre-image and the image of a point after dilation?

A: The pre-image and the image of a point after dilation are related by the scale factor of dilation. The pre-image is the original point, while the image is the dilated point.

Conclusion

In conclusion, dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation, which means that the dilated figure is similar to the original figure. The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated figure to the original figure. The dilated triangle has several properties that are important to understand, including similarity, scale factor, and center of dilation. Dilation has several real-world applications, including architecture, engineering, and art.

Frequently Asked Questions

Q: What is dilation? A: Dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation, which means that the dilated figure is similar to the original figure.
Q: What is the center of dilation? A: The center of dilation is the point around which the dilation takes place. In the given problem, the center of dilation is at the origin (0, 0).
Q: What is the scale factor of dilation? A: The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated figure to the original figure.
Q: How do I find the pre-image of a point after dilation? A: To find the pre-image of a point after dilation, you need to use the given vertices of the dilated triangle. You can use the fact that the center of dilation is at the origin (0, 0) to find the pre-image of the point.
Q: What are the properties of the dilated triangle? A: The dilated triangle has several properties that are important to understand:
- Similarity: The dilated triangle is similar to the original triangle, meaning that the corresponding sides are proportional.
- Scale factor: The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated triangle to the original triangle.
- Center of dilation: The center of dilation is the point around which the dilation takes place. In the given problem, the center of dilation is at the origin (0, 0).
Q: What are the real-world applications of dilation? A: Dilation has several real-world applications, including:
- Architecture: Dilation is used in architecture to create larger or smaller versions of buildings and other structures.
- Engineering: Dilation is used in engineering to create larger or smaller versions of machines and other devices.
- Art: Dilation is used in art to create larger or smaller versions of images and other creative works.
Q: How do I find the pre-image of point C' after dilation? A: To find the pre-image of point C' after dilation, you need to use the given vertices of the dilated triangle. You can use the fact that the center of dilation is at the origin (0, 0) to find the pre-image of point C'.
Q: What is the relationship between the pre-image and the image of a point after dilation? A: The pre-image and the image of a point after dilation are related by the scale factor of dilation. The pre-image is the original point, while the image is the dilated point.

Conclusion

In conclusion, dilation is a transformation that changes the size of a figure, but not its shape. It is a type of similarity transformation, which means that the dilated figure is similar to the original figure. The scale factor of dilation is the ratio of the lengths of the corresponding sides of the dilated figure to the original figure. The dilated triangle has several properties that are important to understand, including similarity, scale factor, and center of dilation. Dilation has several real-world applications, including architecture, engineering, and art.