At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To fully understand the relationship between the pre-image and the image under a dilation with a scale factor of [tex]\(\frac{1}{5}\)[/tex], let's break down the properties of dilation in a geometrical context.
### Properties of Dilation:
1. Scale Factor:
- The scale factor given is [tex]\(\frac{1}{5}\)[/tex].
- This indicates that the image will be scaled down to one-fifth of the original size of the pre-image.
2. Length Relationships:
- All lengths in the image will be [tex]\(\frac{1}{5}\)[/tex] of the corresponding lengths in the pre-image.
- So, if a side of the pre-image is 5 units long, the corresponding side in the image will be [tex]\(5 \times \frac{1}{5} = 1\)[/tex] unit long.
3. Angle Preservation:
- The angles in the pre-image and the image will remain congruent.
- This means that the measure of each angle in the pre-image is exactly the same in the image.
4. Similarity:
- The pre-image and the image are similar figures.
- Similar figures have the same shape but different sizes. All corresponding angles are the same and all corresponding sides are proportional to each other by the scale factor.
### Quadrant Consideration:
- The problem states that the pre-image is located in the third quadrant.
- Since dilation is a transformation that affects the size but not the orientation or location with respect to the coordinate system, the image will also be located in the third quadrant.
### Conclusion:
1. Scale Factor: The lengths of the image are [tex]\(\frac{1}{5}\)[/tex] of the lengths of the pre-image.
2. Size Reduction: The image is reduced in size by a factor of [tex]\(\frac{1}{5}\)[/tex].
3. Angle Conservation: The angle measures in the image are the same as those in the pre-image, preserving the shapes' similarities.
4. Quadrant Location: Both the pre-image and the image are located in the third quadrant.
By consolidating all these points, we understand that the dilation reduces the size of the pre-image by the factor of [tex]\(\frac{1}{5}\)[/tex] while preserving angle measures and maintaining the figures' similarity, and both the pre-image and the image reside in the same quadrant, i.e., the third quadrant.
### Properties of Dilation:
1. Scale Factor:
- The scale factor given is [tex]\(\frac{1}{5}\)[/tex].
- This indicates that the image will be scaled down to one-fifth of the original size of the pre-image.
2. Length Relationships:
- All lengths in the image will be [tex]\(\frac{1}{5}\)[/tex] of the corresponding lengths in the pre-image.
- So, if a side of the pre-image is 5 units long, the corresponding side in the image will be [tex]\(5 \times \frac{1}{5} = 1\)[/tex] unit long.
3. Angle Preservation:
- The angles in the pre-image and the image will remain congruent.
- This means that the measure of each angle in the pre-image is exactly the same in the image.
4. Similarity:
- The pre-image and the image are similar figures.
- Similar figures have the same shape but different sizes. All corresponding angles are the same and all corresponding sides are proportional to each other by the scale factor.
### Quadrant Consideration:
- The problem states that the pre-image is located in the third quadrant.
- Since dilation is a transformation that affects the size but not the orientation or location with respect to the coordinate system, the image will also be located in the third quadrant.
### Conclusion:
1. Scale Factor: The lengths of the image are [tex]\(\frac{1}{5}\)[/tex] of the lengths of the pre-image.
2. Size Reduction: The image is reduced in size by a factor of [tex]\(\frac{1}{5}\)[/tex].
3. Angle Conservation: The angle measures in the image are the same as those in the pre-image, preserving the shapes' similarities.
4. Quadrant Location: Both the pre-image and the image are located in the third quadrant.
By consolidating all these points, we understand that the dilation reduces the size of the pre-image by the factor of [tex]\(\frac{1}{5}\)[/tex] while preserving angle measures and maintaining the figures' similarity, and both the pre-image and the image reside in the same quadrant, i.e., the third quadrant.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.