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.
