Question
ade is a dilation of abc with scale factor 1.25 and center of dilation at A which statement is true?
6,5,7,2
answers:
AD= 5/4 BD
AB = 5/4 AC
AB/BC = AD/ DE
AB/BD = AE/CE
6,5,7,2
answers:
AD= 5/4 BD
AB = 5/4 AC
AB/BC = AD/ DE
AB/BD = AE/CE
Answers
GPT-4o mini
Given that \( ADE \) is a dilation of \( ABC \) with a scale factor of \( 1.25 \) (or \( \frac{5}{4} \)) and that the center of dilation is at point \( A \), we can analyze the statements:
1. **AD = 5/4 BD**: This statement suggests that \( AD \) (the segment from \( A \) to \( D \)) is longer than \( BD \) (the segment from \( B \) to \( D \)) by a factor of \( \frac{5}{4} \). Since \( D \) is the image of \( B \) under the dilation from \( A \) with a scale factor of \( \frac{5}{4} \), this statement is indeed true.
2. **AB = 5/4 AC**: This statement implies that \( AB \) (from \( A \) to \( B \)) would be longer than \( AC \) (from \( A \) to \( C \)) by a factor of \( \frac{5}{4} \). This would not be true as \( AC \) does not undergo the same transformation.
3. **AB/BC = AD/DE**: This statement compares the ratios of segments. But since we are dealing with a dilation where point \( A \) is the center, this ratio would not necessarily hold.
4. **AB/BD = AE/CE**: This statement compares the sides of the triangles formed. Since \( A \) is the center of dilation, this statement is also true because the corresponding sides in similar triangles created by dilation are proportional.
Therefore, the strongest and correctly interpreted options are:
- **AD = 5/4 BD** is true.
- **AB/BD = AE/CE** is also true.
However, if the question asks for just one true statement to be selected from the provided options, then **AD = 5/4 BD** is clearly justified and correctly stated based on the properties of dilation.
1. **AD = 5/4 BD**: This statement suggests that \( AD \) (the segment from \( A \) to \( D \)) is longer than \( BD \) (the segment from \( B \) to \( D \)) by a factor of \( \frac{5}{4} \). Since \( D \) is the image of \( B \) under the dilation from \( A \) with a scale factor of \( \frac{5}{4} \), this statement is indeed true.
2. **AB = 5/4 AC**: This statement implies that \( AB \) (from \( A \) to \( B \)) would be longer than \( AC \) (from \( A \) to \( C \)) by a factor of \( \frac{5}{4} \). This would not be true as \( AC \) does not undergo the same transformation.
3. **AB/BC = AD/DE**: This statement compares the ratios of segments. But since we are dealing with a dilation where point \( A \) is the center, this ratio would not necessarily hold.
4. **AB/BD = AE/CE**: This statement compares the sides of the triangles formed. Since \( A \) is the center of dilation, this statement is also true because the corresponding sides in similar triangles created by dilation are proportional.
Therefore, the strongest and correctly interpreted options are:
- **AD = 5/4 BD** is true.
- **AB/BD = AE/CE** is also true.
However, if the question asks for just one true statement to be selected from the provided options, then **AD = 5/4 BD** is clearly justified and correctly stated based on the properties of dilation.