Answer :
To determine which proportion verifies that [tex]$\triangle ABC$[/tex] and [tex]$\triangle XYZ$[/tex] are similar given that [tex]$\triangle XYZ$[/tex] is a dilation of [tex]$\triangle ABC$[/tex] by a scale factor of 5, we must understand the significance of similarity and dilation in geometry.
Similarity Criteria:
Two triangles are similar if their corresponding angles are equal and the ratios of the lengths of their corresponding sides are equal.
Dilation with a Scale Factor:
When a triangle undergoes a dilation with a scale factor [tex]\( k \)[/tex], each side of the new triangle is [tex]\( k \)[/tex] times as long as the corresponding side of the original triangle.
Given that [tex]$\triangle XYZ$[/tex] is a dilation of [tex]$\triangle ABC$[/tex] with a scale factor of 5, this implies that:
[tex]\[ XY = 5 \cdot AB \\ YZ = 5 \cdot BC \\ XZ = 5 \cdot AC \][/tex]
To verify that the triangles are similar based on the proportions provided, we need to find the proportion that correctly relates the corresponding sides considering the scale factor of 5.
Let's analyze each option:
Option A:
[tex]\[ \frac{AB}{AC} = \frac{XZ}{XY} \][/tex]
This can be written as:
[tex]\[ \frac{AB}{AC} = \frac{5 \cdot AC}{5 \cdot AB} \][/tex]
Simplifying,
[tex]\[ \frac{AB}{AC} = \frac{1}{1} \][/tex]
This is incorrect because the proportion must reflect the scale factor between corresponding sides, not just a ratio of 1.
Option B:
[tex]\[ \frac{BC}{YZ} = \frac{AB}{XZ} \][/tex]
Given the side-lengths are multiplied by the scale factor,
[tex]\[ \frac{BC}{5 \cdot BC} = \frac{AB}{5 \cdot AC} \][/tex]
Simplifying,
[tex]\[ \frac{1}{5} = \frac{1}{5} \][/tex]
This appears to show correct equality in terms of the scale factor, but we should evaluate other options for completeness.
Option C:
[tex]\[ \frac{YZ}{BC} = \frac{AC}{XZ} \][/tex]
Given the side-lengths are multiplied by the scale factor,
[tex]\[ \frac{5 \cdot BC}{BC} = \frac{AC}{5 \cdot AC} \][/tex]
Simplifying,
[tex]\[ 5 = \frac{1}{5} \][/tex]
This is incorrect.
Option D:
[tex]\[ \frac{AB}{XY} = \frac{AC}{XZ} \][/tex]
Given the side-lengths are multiplied by the scale factor,
[tex]\[ \frac{AB}{5 \cdot AB} = \frac{AC}{5 \cdot AC} \][/tex]
Simplifying,
[tex]\[ \frac{1}{5} = \frac{1}{5} \][/tex]
This is correct, as it appropriately considers the scale factor of 5 while maintaining the equality of ratios.
Conclusion:
The correct proportion that verifies [tex]$\triangle ABC$[/tex] and [tex]$\triangle XYZ$[/tex] are similar, given the dilation by a scale factor of 5, is:
[tex]\[ \frac{AB}{XY} = \frac{AC}{XZ} \][/tex]
Thus, the correct answer is:
D.
Similarity Criteria:
Two triangles are similar if their corresponding angles are equal and the ratios of the lengths of their corresponding sides are equal.
Dilation with a Scale Factor:
When a triangle undergoes a dilation with a scale factor [tex]\( k \)[/tex], each side of the new triangle is [tex]\( k \)[/tex] times as long as the corresponding side of the original triangle.
Given that [tex]$\triangle XYZ$[/tex] is a dilation of [tex]$\triangle ABC$[/tex] with a scale factor of 5, this implies that:
[tex]\[ XY = 5 \cdot AB \\ YZ = 5 \cdot BC \\ XZ = 5 \cdot AC \][/tex]
To verify that the triangles are similar based on the proportions provided, we need to find the proportion that correctly relates the corresponding sides considering the scale factor of 5.
Let's analyze each option:
Option A:
[tex]\[ \frac{AB}{AC} = \frac{XZ}{XY} \][/tex]
This can be written as:
[tex]\[ \frac{AB}{AC} = \frac{5 \cdot AC}{5 \cdot AB} \][/tex]
Simplifying,
[tex]\[ \frac{AB}{AC} = \frac{1}{1} \][/tex]
This is incorrect because the proportion must reflect the scale factor between corresponding sides, not just a ratio of 1.
Option B:
[tex]\[ \frac{BC}{YZ} = \frac{AB}{XZ} \][/tex]
Given the side-lengths are multiplied by the scale factor,
[tex]\[ \frac{BC}{5 \cdot BC} = \frac{AB}{5 \cdot AC} \][/tex]
Simplifying,
[tex]\[ \frac{1}{5} = \frac{1}{5} \][/tex]
This appears to show correct equality in terms of the scale factor, but we should evaluate other options for completeness.
Option C:
[tex]\[ \frac{YZ}{BC} = \frac{AC}{XZ} \][/tex]
Given the side-lengths are multiplied by the scale factor,
[tex]\[ \frac{5 \cdot BC}{BC} = \frac{AC}{5 \cdot AC} \][/tex]
Simplifying,
[tex]\[ 5 = \frac{1}{5} \][/tex]
This is incorrect.
Option D:
[tex]\[ \frac{AB}{XY} = \frac{AC}{XZ} \][/tex]
Given the side-lengths are multiplied by the scale factor,
[tex]\[ \frac{AB}{5 \cdot AB} = \frac{AC}{5 \cdot AC} \][/tex]
Simplifying,
[tex]\[ \frac{1}{5} = \frac{1}{5} \][/tex]
This is correct, as it appropriately considers the scale factor of 5 while maintaining the equality of ratios.
Conclusion:
The correct proportion that verifies [tex]$\triangle ABC$[/tex] and [tex]$\triangle XYZ$[/tex] are similar, given the dilation by a scale factor of 5, is:
[tex]\[ \frac{AB}{XY} = \frac{AC}{XZ} \][/tex]
Thus, the correct answer is:
D.
