The Composition \[$ D_{O, 0.75}(x, Y) \cdot D_{O, 2}(x, Y) \$\] Is Applied To \[$\triangle LMN\$\] To Create \[$\triangle L''M''N''\$\].Which Statements Must Be True Regarding The Two Triangles? Check All That Apply.-

Introduction

In geometry, dilations are transformations that change the size of a figure. When two dilations are composed, the resulting transformation can have a significant impact on the original figure. In this article, we will explore the composition of two dilations, specifically the composition of $D_{O, 0.75}(x, y) \cdot D_{O, 2}(x, y)$, and examine the resulting relationship between the original triangle $\triangle LMN$ and the transformed triangle $\triangle L''M''N''$.

Understanding Dilations

A dilation is a transformation that changes the size of a figure by a scale factor. The scale factor is a ratio that determines the size of the transformed figure. In the case of the composition $D_{O, 0.75}(x, y) \cdot D_{O, 2}(x, y)$, we have two dilations with different scale factors: 0.75 and 2.

The Composition of Dilations

When two dilations are composed, the resulting transformation is also a dilation. The scale factor of the resulting dilation is the product of the scale factors of the two individual dilations. In this case, the scale factor of the resulting dilation is $0.75 \cdot 2 = 1.5$.

Applying the Composition to the Triangle

Now that we have understood the composition of the two dilations, let's apply it to the original triangle $\triangle LMN$. The composition $D_{O, 0.75}(x, y) \cdot D_{O, 2}(x, y)$ is applied to the triangle, resulting in the transformed triangle $\triangle L''M''N''$.

Properties of the Transformed Triangle

The transformed triangle $\triangle L''M''N''$ has several properties that are related to the original triangle $\triangle LMN$. Some of these properties are:

1. Similarity

The transformed triangle $\triangle L''M''N''$ is similar to the original triangle $\triangle LMN$. This means that the corresponding angles of the two triangles are equal, and the corresponding sides are proportional.

2. Scale Factor

The scale factor of the transformed triangle $\triangle L''M''N''$ is 1.5, which is the product of the scale factors of the two individual dilations.

3. Center of Dilation

The center of dilation for the composition $D_{O, 0.75}(x, y) \cdot D_{O, 2}(x, y)$ is the same as the center of dilation for the individual dilations, which is point O.

4. Distance from the Center

The distance from the center of dilation O to the vertices of the transformed triangle $\triangle L''M''N''$ is 1.5 times the distance from the center of dilation O to the vertices of the original triangle $\triangle LMN$.

5. Angle Measures

The angle measures of the transformed triangle $\triangle L''M''N''$ are the same as the angle measures of the original triangle $\triangle LMN$.

6. Side Lengths

The side lengths of the transformed triangle $\triangle L''M''N''$ are 1.5 times the side lengths of the original triangle $\triangle LMN$.

Conclusion

In conclusion, the composition of the two dilations $D_{O, 0.75}(x, y) \cdot D_{O, 2}(x, y)$ results in a transformed triangle $\triangle L''M''N''$ that is similar to the original triangle $\triangle LMN$. The transformed triangle has a scale factor of 1.5, and its properties are related to the original triangle. Understanding the composition of dilations is essential in geometry, as it helps us analyze and solve problems involving transformations.

Key Takeaways

• The composition of two dilations results in a transformed figure that is similar to the original figure.
• The scale factor of the resulting dilation is the product of the scale factors of the two individual dilations.
• The center of dilation for the composition is the same as the center of dilation for the individual dilations.
• The distance from the center of dilation to the vertices of the transformed figure is the product of the scale factors of the two individual dilations.
• The angle measures of the transformed figure are the same as the angle measures of the original figure.
• The side lengths of the transformed figure are the product of the scale factors of the two individual dilations.

Practice Problems

1. Apply the composition $D_{O, 0.75}(x, y) \cdot D_{O, 2}(x, y)$ to the triangle $\triangle ABC$ and determine the properties of the transformed triangle $\triangle A''B''C''$.
2. Find the scale factor of the transformed triangle $\triangle DEF$ after applying the composition $D_{O, 0.5}(x, y) \cdot D_{O, 3}(x, y)$.
3. Determine the center of dilation for the composition $D_{O, 0.75}(x, y) \cdot D_{O, 2}(x, y)$ and explain why it is the same as the center of dilation for the individual dilations.

References

• [1] Geometry: A Comprehensive Course, 3rd Edition, by Dan Pedoe
• [2] Mathematics: A Human Approach, 2nd Edition, by Harold R. Jacobs
• [3] Geometry: A Modern Approach, 2nd Edition, by David C. Kay

Frequently Asked Questions

Q: What is the composition of dilations?

A: The composition of dilations is a transformation that combines two or more dilations to create a new dilation. The resulting dilation has a scale factor that is the product of the scale factors of the individual dilations.

Q: How do I apply the composition of dilations to a triangle?

A: To apply the composition of dilations to a triangle, you need to identify the center of dilation, the scale factors of the individual dilations, and the vertices of the triangle. Then, you can use the formula for the composition of dilations to find the new vertices of the transformed triangle.

Q: What are the properties of the transformed triangle after applying the composition of dilations?

A: The transformed triangle has several properties that are related to the original triangle. These properties include:

• Similarity: The transformed triangle is similar to the original triangle.
• Scale factor: The scale factor of the transformed triangle is the product of the scale factors of the individual dilations.
• Center of dilation: The center of dilation for the composition is the same as the center of dilation for the individual dilations.
• Distance from the center: The distance from the center of dilation to the vertices of the transformed triangle is the product of the scale factors of the individual dilations.
• Angle measures: The angle measures of the transformed triangle are the same as the angle measures of the original triangle.
• Side lengths: The side lengths of the transformed triangle are the product of the scale factors of the individual dilations.

Q: How do I find the scale factor of the transformed triangle after applying the composition of dilations?

A: To find the scale factor of the transformed triangle, you need to multiply the scale factors of the individual dilations. For example, if the scale factors of the individual dilations are 0.75 and 2, the scale factor of the transformed triangle is 0.75 x 2 = 1.5.

Q: What is the center of dilation for the composition of dilations?

A: The center of dilation for the composition of dilations is the same as the center of dilation for the individual dilations. This means that the center of dilation remains the same even after applying the composition of dilations.

Q: How do I determine the distance from the center of dilation to the vertices of the transformed triangle?

A: To determine the distance from the center of dilation to the vertices of the transformed triangle, you need to multiply the distance from the center of dilation to the vertices of the original triangle by the scale factor of the transformed triangle.

Q: What are the angle measures of the transformed triangle after applying the composition of dilations?

A: The angle measures of the transformed triangle are the same as the angle measures of the original triangle. This means that the composition of dilations does not change the angle measures of the triangle.

Q: How do I find the side lengths of the transformed triangle after applying the composition of dilations?

A: To find the side lengths of the transformed triangle, you need to multiply the side lengths of the original triangle by the scale factor of the transformed triangle.

Common Mistakes

• Failing to identify the center of dilation and the scale factors of the individual dilations.
• Not multiplying the scale factors of the individual dilations to find the scale factor of the transformed triangle.
• Not using the correct formula for the composition of dilations.
• Not considering the properties of the transformed triangle, such as similarity and scale factor.

Tips and Tricks

• Make sure to identify the center of dilation and the scale factors of the individual dilations before applying the composition of dilations.
• Use the formula for the composition of dilations to find the new vertices of the transformed triangle.
• Consider the properties of the transformed triangle, such as similarity and scale factor, to ensure that you are applying the composition of dilations correctly.
• Practice applying the composition of dilations to different triangles to become more comfortable with the concept.

Practice Problems

1. Apply the composition of dilations to the triangle $\triangle ABC$ with a center of dilation at point O and scale factors of 0.75 and 2. Find the new vertices of the transformed triangle and determine its properties.
2. Find the scale factor of the transformed triangle after applying the composition of dilations to the triangle $\triangle DEF$ with a center of dilation at point O and scale factors of 0.5 and 3.
3. Determine the center of dilation for the composition of dilations to the triangle $\triangle GHI$ with a center of dilation at point O and scale factors of 0.75 and 2.

References

• [1] Geometry: A Comprehensive Course, 3rd Edition, by Dan Pedoe
• [2] Mathematics: A Human Approach, 2nd Edition, by Harold R. Jacobs
• [3] Geometry: A Modern Approach, 2nd Edition, by David C. Kay

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources on the topic.