Dilating a Triangle with a Scale Factor of 2

Oct 13, 2025 by windailys.com 36 views

Introduction

Dilation is a fundamental concept in geometry that involves changing the size of a figure while maintaining its shape. When a figure is dilated, its size is increased or decreased by a certain scale factor, and the center of dilation remains fixed. In this article, we will explore the relationship between an original triangle and its image when dilated with a scale factor of 2 using a fixed center of dilation.

Understanding Dilation

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 certain scale factor, which is a ratio of the lengths of the corresponding sides of the original and image figures. The scale factor is always greater than 0, and it can be greater than 1 (enlargement) or less than 1 (reduction).

Dilating a Triangle with a Scale Factor of 2

When a triangle is dilated with a scale factor of 2, its size is doubled. This means that the lengths of the corresponding sides of the original and image triangles are in a ratio of 2:1. The image triangle is larger than the original triangle, and its vertices are farther apart.

Relationship Between the Original and Image Triangles

Now, let's examine the relationship between the original triangle and its image when dilated with a scale factor of 2. We will consider the following statements:

A. The corresponding sides are congruent.
B. The image is smaller than the original triangle.
C. The image is larger than the original triangle.
D. The image is congruent to the original triangle.

Analyzing Statement A

Statement A claims that the corresponding sides of the original and image triangles are congruent. However, this is not true when the scale factor is 2. The lengths of the corresponding sides of the original and image triangles are in a ratio of 2:1, which means that the image triangle has longer sides than the original triangle.

Analyzing Statement B

Statement B claims that the image is smaller than the original triangle. However, this is not true when the scale factor is 2. The image triangle is larger than the original triangle, not smaller.

Analyzing Statement C

Statement C claims that the image is larger than the original triangle. This is true when the scale factor is 2. The lengths of the corresponding sides of the original and image triangles are in a ratio of 2:1, which means that the image triangle has longer sides than the original triangle.

Analyzing Statement D

Statement D claims that the image is congruent to the original triangle. However, this is not true when the scale factor is 2. The image triangle is larger than the original triangle, and its vertices are farther apart.

Conclusion

In conclusion, when a triangle is dilated with a scale factor of 2 using a fixed center of dilation, the corresponding sides of the original and image triangles are not congruent. The image triangle is larger than the original triangle, and its vertices are farther apart. Therefore, the correct statement is:

C. The image is larger than the original triangle.

This statement accurately describes the relationship between the original triangle and its image when dilated with a scale factor of 2.

Frequently Asked Questions

Q: What is dilation in geometry?

A: 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 certain scale factor.

Q: What is a scale factor?

A: A scale factor is a ratio of the lengths of the corresponding sides of the original and image figures.

Q: What happens to the size of a triangle when it is dilated with a scale factor of 2?

A: The size of the triangle is doubled.

Q: Are the corresponding sides of the original and image triangles congruent when the scale factor is 2?

A: No, the corresponding sides of the original and image triangles are not congruent when the scale factor is 2.

Q: Is the image triangle smaller than the original triangle when the scale factor is 2?

A: No, the image triangle is larger than the original triangle when the scale factor is 2.

Q: Is the image triangle congruent to the original triangle when the scale factor is 2?

A: No, the image triangle is not congruent to the original triangle when the scale factor is 2.

Key Takeaways

Dilation is a transformation that changes the size of a figure, but not its shape.
A scale factor is a ratio of the lengths of the corresponding sides of the original and image figures.
When a triangle is dilated with a scale factor of 2, its size is doubled.
The corresponding sides of the original and image triangles are not congruent when the scale factor is 2.
The image triangle is larger than the original triangle when the scale factor is 2.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Dilation" by Math Open Reference
[3] "Scale Factor" by Math Is Fun

Introduction

Understanding Dilation

Dilating a Triangle with a Scale Factor of 2

Relationship Between the Original and Image Triangles

Now, let's examine the relationship between the original triangle and its image when dilated with a scale factor of 2. We will consider the following statements:

A. The corresponding sides are congruent.
B. The image is smaller than the original triangle.
C. The image is larger than the original triangle.
D. The image is congruent to the original triangle.

Analyzing Statement A

Analyzing Statement B

Statement B claims that the image is smaller than the original triangle. However, this is not true when the scale factor is 2. The image triangle is larger than the original triangle, not smaller.

Analyzing Statement C

Analyzing Statement D

Conclusion

C. The image is larger than the original triangle.

This statement accurately describes the relationship between the original triangle and its image when dilated with a scale factor of 2.

Frequently Asked Questions

Q: What is dilation in geometry?

A: 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 certain scale factor.

Q: What is a scale factor?

A: A scale factor is a ratio of the lengths of the corresponding sides of the original and image figures.

Q: What happens to the size of a triangle when it is dilated with a scale factor of 2?

A: The size of the triangle is doubled.

Q: Are the corresponding sides of the original and image triangles congruent when the scale factor is 2?

A: No, the corresponding sides of the original and image triangles are not congruent when the scale factor is 2.

Q: Is the image triangle smaller than the original triangle when the scale factor is 2?

A: No, the image triangle is larger than the original triangle when the scale factor is 2.

Q: Is the image triangle congruent to the original triangle when the scale factor is 2?

A: No, the image triangle is not congruent to the original triangle when the scale factor is 2.

Q&A Session

Q: What is the effect of dilation on the size of a figure?

A: Dilation changes the size of a figure, but not its shape. When a figure is dilated, its size is increased or decreased by a certain scale factor.

Q: What is the relationship between the original and image triangles when the scale factor is 2?

A: The image triangle is larger than the original triangle, and its vertices are farther apart.

Q: Can the scale factor be greater than 1?

A: Yes, the scale factor can be greater than 1, which means that the size of the figure is increased.

Q: Can the scale factor be less than 1?

A: Yes, the scale factor can be less than 1, which means that the size of the figure is decreased.

Q: What is the effect of dilation on the shape of a figure?

A: Dilation does not change the shape of a figure. The shape remains the same, but the size is changed.

Q: Can a figure be dilated with a scale factor of 0?

A: No, a figure cannot be dilated with a scale factor of 0. The scale factor must be greater than 0.

Q: Can a figure be dilated with a scale factor of infinity?

A: No, a figure cannot be dilated with a scale factor of infinity. The scale factor must be a finite number.

Key Takeaways

Dilation is a transformation that changes the size of a figure, but not its shape.
A scale factor is a ratio of the lengths of the corresponding sides of the original and image figures.
When a triangle is dilated with a scale factor of 2, its size is doubled.
The corresponding sides of the original and image triangles are not congruent when the scale factor is 2.
The image triangle is larger than the original triangle when the scale factor is 2.

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Dilation" by Math Open Reference
[3] "Scale Factor" by Math Is Fun