Dilation by a Scale Factor of 3

Oct 13, 2025 by windailys.com 112 views

Introduction

In geometry, transformations are used to change the position, size, or shape of a figure. Two common transformations are reflection and dilation. In this article, we will explore the effects of reflecting a triangle over the x-axis and then dilating it by a scale factor of 3 about the origin.

Reflection over the x-axis

When a figure is reflected over the x-axis, its y-coordinates change sign. This means that the x-coordinates remain the same, but the y-coordinates become their negative counterparts.

Example: Reflection of a Point

Suppose we have a point (2, 3) and we reflect it over the x-axis. The new coordinates of the point will be (2, -3).

Reflection of a Triangle

Now, let's consider a triangle ABC with vertices A(1, 2), B(3, 4), and C(2, 1). When we reflect this triangle over the x-axis, the new coordinates of the vertices will be:

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

Dilation by a Scale Factor of 3

A dilation is a transformation that changes the size of a figure. When a figure is dilated by a scale factor of k, its coordinates are multiplied by k. In this case, we will dilate the reflected triangle by a scale factor of 3.

Example: Dilation of a Point

Suppose we have a point (2, -3) and we dilate it by a scale factor of 3. The new coordinates of the point will be (6, -9).

Dilation of a Triangle

Now, let's consider the reflected triangle A'B'C' with vertices A'(1, -2), B'(3, -4), and C'(2, -1). When we dilate this triangle by a scale factor of 3, the new coordinates of the vertices will be:

A''(3, -6)
B''(9, -12)
C''(6, -3)

Conclusion

In conclusion, when a triangle is reflected over the x-axis and then dilated by a scale factor of 3 about the origin, the vertices of the new triangle are located at A''(3, -6), B''(9, -12), and C''(6, -3).

Discussion

What happens when a figure is reflected over the y-axis?
How does the scale factor affect the size of the figure?
Can you think of any real-world applications of reflection and dilation?

Answer Key

A. (6, 6), (2, -4), (4, -2)

Additional Resources

Khan Academy: Reflection and Dilation
Math Open Reference: Reflection and Dilation
Geometry Transformations: Reflection and Dilation (PDF)
Geometry Transformations: Reflection and Dilation Q&A =====================================================

Introduction

In our previous article, we explored the effects of reflecting a triangle over the x-axis and then dilating it by a scale factor of 3 about the origin. In this article, we will answer some common questions related to geometry transformations, reflection, and dilation.

Q&A

Q: What is the difference between reflection and dilation?

A: Reflection is a transformation that changes the position of a figure, while dilation is a transformation that changes the size of a figure.

Q: How do you reflect a figure over the x-axis?

A: To reflect a figure over the x-axis, you change the sign of the y-coordinates of the figure.

Q: How do you dilate a figure by a scale factor of k?

A: To dilate a figure by a scale factor of k, you multiply the coordinates of the figure by k.

Q: What happens when a figure is reflected over the y-axis?

A: When a figure is reflected over the y-axis, the x-coordinates change sign, while the y-coordinates remain the same.

Q: Can you think of any real-world applications of reflection and dilation?

A: Yes, reflection and dilation have many real-world applications, such as:

Designing buildings and bridges
Creating art and graphics
Understanding the behavior of objects in motion
Solving problems in physics and engineering

Q: How do you determine the scale factor of a dilation?

A: The scale factor of a dilation is the ratio of the new size of the figure to the original size of the figure.

Q: Can you dilate a figure by a negative scale factor?

A: Yes, you can dilate a figure by a negative scale factor, but it will result in a reflection of the figure.

Q: How do you determine the coordinates of a figure after a dilation?

A: To determine the coordinates of a figure after a dilation, you multiply the original coordinates of the figure by the scale factor.

Q: Can you dilate a figure by a scale factor of 0?

A: No, you cannot dilate a figure by a scale factor of 0, as it would result in a figure with zero size.

Conclusion

In conclusion, reflection and dilation are two important transformations in geometry that can be used to change the position and size of a figure. By understanding these transformations, you can solve problems in various fields, such as art, design, physics, and engineering.

Discussion

What are some other types of transformations in geometry?
How do you determine the coordinates of a figure after a reflection?
Can you think of any other real-world applications of reflection and dilation?

Answer Key

Reflection: changing the sign of the y-coordinates
Dilation: multiplying the coordinates by a scale factor
Scale factor: the ratio of the new size to the original size
Negative scale factor: a reflection of the figure
Coordinates after dilation: multiply the original coordinates by the scale factor

Additional Resources

Khan Academy: Reflection and Dilation
Math Open Reference: Reflection and Dilation
Geometry Transformations: Reflection and Dilation (PDF)