A Dilation Produces A Smaller Figure. Which Is A Possible Scale Factor?A. $-2$ B. $\frac{1}{3}$ C. $1$ D. $4$

by ADMIN 115 views

Introduction

In mathematics, a dilation is a transformation that changes the size of a figure. When a dilation produces a smaller figure, it means that the new figure is smaller than the original one. In this article, we will explore the concept of scale factors and determine which of the given options is a possible scale factor for a dilation that produces a smaller figure.

Understanding Scale Factors

A scale factor is a number that represents the ratio of the size of the new figure to the size of the original figure. It is a measure of how much the figure has been enlarged or reduced. When a dilation produces a smaller figure, the scale factor is less than 1. This means that the new figure is smaller than the original one.

Possible Scale Factors for a Smaller Figure

Now, let's examine the given options and determine which one is a possible scale factor for a dilation that produces a smaller figure.

Option A: −2-2

A scale factor of −2-2 would mean that the new figure is twice as large as the original figure, but with the opposite orientation. This is not a possible scale factor for a dilation that produces a smaller figure, as it would actually enlarge the figure.

Option B: 13\frac{1}{3}

A scale factor of 13\frac{1}{3} would mean that the new figure is one-third the size of the original figure. This is a possible scale factor for a dilation that produces a smaller figure, as it would reduce the size of the figure.

Option C: 11

A scale factor of 11 would mean that the new figure is the same size as the original figure. This is not a possible scale factor for a dilation that produces a smaller figure, as it would not change the size of the figure.

Option D: 44

A scale factor of 44 would mean that the new figure is four times as large as the original figure. This is not a possible scale factor for a dilation that produces a smaller figure, as it would actually enlarge the figure.

Conclusion

In conclusion, the only possible scale factor for a dilation that produces a smaller figure is 13\frac{1}{3}. This is because a scale factor of 13\frac{1}{3} would reduce the size of the figure, making it smaller than the original one.

Examples and Applications

Here are some examples and applications of dilations with scale factors:

  • Example 1: A dilation with a scale factor of 12\frac{1}{2} is applied to a square with a side length of 4 units. What is the area of the new square?
  • Example 2: A dilation with a scale factor of 34\frac{3}{4} is applied to a triangle with a base of 6 units and a height of 8 units. What is the area of the new triangle?
  • Application: In architecture, dilations are used to design buildings and structures. A scale factor of 110\frac{1}{10} might be used to create a model of a building that is one-tenth the size of the actual building.

Tips and Tricks

Here are some tips and tricks for working with dilations and scale factors:

  • Tip 1: When working with dilations, it's essential to understand the concept of scale factors and how they affect the size of the figure.
  • Tip 2: To determine the scale factor of a dilation, divide the size of the new figure by the size of the original figure.
  • Tip 3: When applying a dilation, make sure to consider the orientation of the figure and the direction of the dilation.

Common Mistakes

Here are some common mistakes to avoid when working with dilations and scale factors:

  • Mistake 1: Failing to understand the concept of scale factors and how they affect the size of the figure.
  • Mistake 2: Not considering the orientation of the figure and the direction of the dilation.
  • Mistake 3: Not using the correct scale factor when applying a dilation.

Final Thoughts

In conclusion, a dilation produces a smaller figure when the scale factor is less than 1. The only possible scale factor for a dilation that produces a smaller figure is 13\frac{1}{3}. By understanding the concept of scale factors and how they affect the size of the figure, you can apply dilations with confidence and accuracy.

Introduction

In our previous article, we explored the concept of dilations and scale factors, and determined that a dilation produces a smaller figure when the scale factor is less than 1. In this article, we will answer some frequently asked questions about dilations and scale factors.

Q&A

Q: What is a dilation?

A: A dilation is a transformation that changes the size of a figure. It can be an enlargement or a reduction, depending on the scale factor.

Q: What is a scale factor?

A: A scale factor is a number that represents the ratio of the size of the new figure to the size of the original figure. It is a measure of how much the figure has been enlarged or reduced.

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

A: To determine the scale factor of a dilation, divide the size of the new figure by the size of the original figure.

Q: What is the difference between a dilation and a translation?

A: A dilation changes the size of a figure, while a translation changes the position of a figure. A dilation can be an enlargement or a reduction, while a translation always moves the figure in a specific direction.

Q: Can a dilation produce a larger figure?

A: Yes, a dilation can produce a larger figure when the scale factor is greater than 1.

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

A: The effect of a dilation on the area of a figure depends on the scale factor. If the scale factor is greater than 1, the area of the figure will increase. If the scale factor is less than 1, the area of the figure will decrease.

Q: Can a dilation be applied to a three-dimensional figure?

A: Yes, a dilation can be applied to a three-dimensional figure. However, the scale factor will affect the size of the figure in all three dimensions.

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

A: To apply a dilation to a figure, multiply the coordinates of each point on the figure by the scale factor.

Q: What is the importance of dilations in real-life applications?

A: Dilations are used in various real-life applications, such as architecture, engineering, and computer graphics. They are also used in art and design to create different perspectives and effects.

Q: Can a dilation be reversed?

A: Yes, a dilation can be reversed by applying the inverse scale factor. This will restore the original figure.

Examples and Applications

Here are some examples and applications of dilations:

  • Example 1: A dilation with a scale factor of 2 is applied to a square with a side length of 4 units. What is the area of the new square?
  • Example 2: A dilation with a scale factor of 1/2 is applied to a triangle with a base of 6 units and a height of 8 units. What is the area of the new triangle?
  • Application: In architecture, dilations are used to design buildings and structures. A scale factor of 1/10 might be used to create a model of a building that is one-tenth the size of the actual building.

Tips and Tricks

Here are some tips and tricks for working with dilations:

  • Tip 1: When working with dilations, it's essential to understand the concept of scale factors and how they affect the size of the figure.
  • Tip 2: To determine the scale factor of a dilation, divide the size of the new figure by the size of the original figure.
  • Tip 3: When applying a dilation, make sure to consider the orientation of the figure and the direction of the dilation.

Common Mistakes

Here are some common mistakes to avoid when working with dilations:

  • Mistake 1: Failing to understand the concept of scale factors and how they affect the size of the figure.
  • Mistake 2: Not considering the orientation of the figure and the direction of the dilation.
  • Mistake 3: Not using the correct scale factor when applying a dilation.

Final Thoughts

In conclusion, dilations are an essential concept in mathematics and have various real-life applications. By understanding the concept of scale factors and how they affect the size of the figure, you can apply dilations with confidence and accuracy.