Under A Dilation Centered At The Origin, The Image Is Congruent To The Preimage. What Is The Scale Factor?A. -1 Or 0 B. 0 Or 1 C. Only 1 D. -1 Or 1
Introduction
In mathematics, dilation is a transformation that changes the size of a figure. When a dilation is centered at the origin, the image is congruent to the preimage, meaning that the two figures have the same shape and size. However, the scale factor, which is a measure of how much the figure is enlarged or reduced, can be either positive or negative. In this article, we will explore the concept of dilation and scale factor, and determine the correct answer to the question: what is the scale factor when the image is congruent to the preimage?
What is Dilation?
Dilation is a transformation that changes the size of a figure. It is a type of similarity transformation that can be represented by a scale factor. When a dilation is centered at the origin, the image is congruent to the preimage, meaning that the two figures have the same shape and size. However, the scale factor can be either positive or negative, depending on whether the figure is enlarged or reduced.
Scale Factor
The scale factor is a measure of how much the figure is enlarged or reduced. It is a ratio of the length of the image to the length of the preimage. When the image is congruent to the preimage, the scale factor is equal to 1. This means that the figure is neither enlarged nor reduced, and the two figures have the same size.
Types of Scale Factors
There are two types of scale factors: positive and negative. A positive scale factor represents an enlargement, while a negative scale factor represents a reduction. When the image is congruent to the preimage, the scale factor is equal to 1, which is a positive value.
Examples of Dilation
Here are some examples of dilation:
- Enlargement: When a figure is enlarged by a scale factor of 2, the image is twice as large as the preimage.
- Reduction: When a figure is reduced by a scale factor of 1/2, the image is half as large as the preimage.
- Congruent: When a figure is congruent to the preimage, the scale factor is equal to 1.
Conclusion
In conclusion, when the image is congruent to the preimage, the scale factor is equal to 1. This means that the figure is neither enlarged nor reduced, and the two figures have the same size. The correct answer to the question is B. 0 or 1.
Final Answer
The final answer is B. 0 or 1.
References
- [1] "Dilation" by Math Open Reference. Retrieved 2023-12-01.
- [2] "Scale Factor" by Math Is Fun. Retrieved 2023-12-01.
Additional Resources
- [1] Khan Academy. "Dilation". Retrieved 2023-12-01.
- [2] Mathway. "Dilation". Retrieved 2023-12-01.
Discussion
What is your understanding of dilation and scale factor? Do you have any questions or comments about this article? Please share your thoughts in the discussion section below.
Discussion Section
Post a Comment
- Comment: I have a question about dilation. Can you explain how to find the scale factor of a figure that has been enlarged by a certain amount?
- Response: To find the scale factor of a figure that has been enlarged by a certain amount, you can use the formula: scale factor = (length of image) / (length of preimage). For example, if the length of the image is 6 and the length of the preimage is 3, the scale factor would be 6/3 = 2.
Post a Comment
- Comment: I have a question about the scale factor. Can you explain why the scale factor is equal to 1 when the image is congruent to the preimage?
- Response: The scale factor is equal to 1 when the image is congruent to the preimage because the two figures have the same size. When the scale factor is equal to 1, the figure is neither enlarged nor reduced, and the two figures have the same size.
Post a Comment
- Comment: I have a question about dilation. Can you explain how to find the image of a figure that has been dilated by a certain amount?
- Response: To find the image of a figure that has been dilated by a certain amount, you can use the formula: image = (scale factor) * (preimage). For example, if the scale factor is 2 and the preimage is a square with a side length of 3, the image would be a square with a side length of 6.
Post a Comment
- Comment: I have a question about the scale factor. Can you explain why the scale factor can be either positive or negative?
- Response: The scale factor can be either positive or negative because it represents an enlargement or reduction. A positive scale factor represents an enlargement, while a negative scale factor represents a reduction.
Dilation and Scale Factor Q&A =============================
Q: What is dilation?
A: Dilation is a transformation that changes the size of a figure. It is a type of similarity transformation that can be represented by a scale factor.
Q: What is the scale factor?
A: The scale factor is a measure of how much the figure is enlarged or reduced. It is a ratio of the length of the image to the length of the preimage.
Q: What is the difference between a positive and negative scale factor?
A: A positive scale factor represents an enlargement, while a negative scale factor represents a reduction.
Q: What is the scale factor when the image is congruent to the preimage?
A: The scale factor is equal to 1 when the image is congruent to the preimage.
Q: How do you find the scale factor of a figure that has been enlarged by a certain amount?
A: To find the scale factor of a figure that has been enlarged by a certain amount, you can use the formula: scale factor = (length of image) / (length of preimage).
Q: How do you find the image of a figure that has been dilated by a certain amount?
A: To find the image of a figure that has been dilated by a certain amount, you can use the formula: image = (scale factor) * (preimage).
Q: Can the scale factor be equal to 0?
A: No, the scale factor cannot be equal to 0. This is because the scale factor is a ratio of the length of the image to the length of the preimage, and a ratio cannot be equal to 0.
Q: Can the scale factor be equal to -1?
A: Yes, the scale factor can be equal to -1. This represents a reduction of the figure by a factor of 1.
Q: Can the scale factor be equal to 1?
A: Yes, the scale factor can be equal to 1. This represents a congruence between the image and the preimage.
Q: What is the relationship between the scale factor and the coordinates of the figure?
A: The scale factor affects the coordinates of the figure. When the scale factor is greater than 1, the coordinates of the figure are multiplied by the scale factor. When the scale factor is less than 1, the coordinates of the figure are divided by the scale factor.
Q: Can the scale factor be a fraction?
A: Yes, the scale factor can be a fraction. For example, a scale factor of 1/2 represents a reduction of the figure by a factor of 1/2.
Q: Can the scale factor be a decimal?
A: Yes, the scale factor can be a decimal. For example, a scale factor of 0.5 represents a reduction of the figure by a factor of 0.5.
Q: What is the effect of a scale factor of 0 on the figure?
A: A scale factor of 0 would result in the figure being transformed into a point.
Q: What is the effect of a scale factor of 1 on the figure?
A: A scale factor of 1 would result in the figure being congruent to the preimage.
Q: What is the effect of a scale factor of -1 on the figure?
A: A scale factor of -1 would result in the figure being reflected across the origin.
Q: Can the scale factor be a complex number?
A: Yes, the scale factor can be a complex number. For example, a scale factor of 2 + 3i represents a dilation of the figure by a factor of 2 + 3i.
Q: Can the scale factor be a matrix?
A: Yes, the scale factor can be a matrix. For example, a scale factor of [[2, 0], [0, 2]] represents a dilation of the figure by a factor of 2 in both the x and y directions.
Q: What is the relationship between the scale factor and the area of the figure?
A: The scale factor affects the area of the figure. When the scale factor is greater than 1, the area of the figure is multiplied by the square of the scale factor. When the scale factor is less than 1, the area of the figure is divided by the square of the scale factor.
Q: Can the scale factor be used to find the perimeter of the figure?
A: Yes, the scale factor can be used to find the perimeter of the figure. When the scale factor is greater than 1, the perimeter of the figure is multiplied by the scale factor. When the scale factor is less than 1, the perimeter of the figure is divided by the scale factor.
Q: Can the scale factor be used to find the volume of the figure?
A: Yes, the scale factor can be used to find the volume of the figure. When the scale factor is greater than 1, the volume of the figure is multiplied by the cube of the scale factor. When the scale factor is less than 1, the volume of the figure is divided by the cube of the scale factor.