Answer :
Given that [tex]\(\triangle XYZ\)[/tex] is a dilation of [tex]\(\triangle ABC\)[/tex] by a scale factor of 5, we want to determine which proportion verifies that [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle XYZ\)[/tex] are similar.
For two triangles to be similar, corresponding sides must be proportional. That is, the ratios of the lengths of corresponding sides must be equal.
Let's examine each option in detail:
Option A: [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex]
For [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle ABC\)[/tex]:
- The side [tex]\(XY\)[/tex] in [tex]\(\triangle XYZ\)[/tex] corresponds to the side [tex]\(AB\)[/tex] in [tex]\(\triangle ABC\)[/tex]
- The side [tex]\(XZ\)[/tex] in [tex]\(\triangle XYZ\)[/tex] corresponds to the side [tex]\(AC\)[/tex] in [tex]\(\triangle ABC\)[/tex]
Since [tex]\(\triangle XYZ\)[/tex] is a dilation of [tex]\(\triangle ABC\)[/tex] by a scale factor of 5:
- [tex]\(XY = 5 \cdot AB\)[/tex]
- [tex]\(XZ = 5 \cdot AC\)[/tex]
Then:
[tex]\[ \frac{AB}{5 \cdot AB} = \frac{AC}{5 \cdot AC} \][/tex]
Both sides simplify to [tex]\(\frac{1}{5}\)[/tex]. However, this does not directly verify the similarity of [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle XYZ\)[/tex].
Option B: [tex]\(\frac{YZ}{BC} = \frac{AC}{XZ}\)[/tex]
For sides [tex]\(YZ\)[/tex] and [tex]\(BC\)[/tex]:
- The side [tex]\(YZ\)[/tex] in [tex]\(\triangle XYZ\)[/tex] corresponds to the side [tex]\(BC\)[/tex] in [tex]\(\triangle ABC\)[/tex]
- Since [tex]\(\triangle XYZ\)[/tex] is a dilation of [tex]\(\triangle ABC\)[/tex] by a scale factor of 5, [tex]\(YZ = 5 \cdot BC\)[/tex]
Then:
[tex]\[ \frac{5 \cdot BC}{BC} = \frac{AC}{5 \cdot AC} \][/tex]
This gives us:
[tex]\[ 5 = \frac{1}{5} \][/tex]
This is not true, so option B does not verify the similarity.
Option C: [tex]\(\frac{BC}{YZ} = \frac{AB}{XZ}\)[/tex]
Considering [tex]\(BC\)[/tex] and [tex]\(YZ\)[/tex]:
- [tex]\(YZ = 5 \cdot BC\)[/tex]
For sides [tex]\(AB\)[/tex] and [tex]\(XZ\)[/tex]:
- [tex]\(XZ = 5 \cdot AC\)[/tex]
Then:
[tex]\[ \frac{BC}{5 \cdot BC} = \frac{AB}{5 \cdot AC} \][/tex]
Both sides simplify to [tex]\(\frac{1}{5}\)[/tex]. While the ratios match, it does not directly verify similarity in a general manner different from the approach we're analyzing.
Option D: [tex]\(\frac{AB}{AC} = \frac{XZ}{XY}\)[/tex]
Considering the ratios of corresponding sides within each triangle:
- [tex]\(\frac{AB}{AC}\)[/tex] is a ratio of sides in [tex]\(\triangle ABC\)[/tex]
- [tex]\(XZ\)[/tex] and [tex]\(XY\)[/tex] are sides corresponding to [tex]\(AC\)[/tex] and [tex]\(AB\)[/tex] respectively in [tex]\(\triangle XYZ\)[/tex]
Since [tex]\(\triangle XYZ\)[/tex] is a dilation of [tex]\(\triangle ABC\)[/tex] by a scale factor of 5:
- [tex]\(XZ = 5 \cdot AC\)[/tex]
- [tex]\(XY = 5 \cdot AB\)[/tex]
Then:
[tex]\[ \frac{AB}{AC} = \frac{5 \cdot AC}{5 \cdot AB} \][/tex]
Both sides simplify to:
[tex]\[ \frac{AB}{AC} \][/tex]
Thus, the ratios of the corresponding sides are equal, verifying that [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle XYZ\)[/tex] are similar.
Therefore, the correct option is:
[tex]\[ \boxed{D} \][/tex]
For two triangles to be similar, corresponding sides must be proportional. That is, the ratios of the lengths of corresponding sides must be equal.
Let's examine each option in detail:
Option A: [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex]
For [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle ABC\)[/tex]:
- The side [tex]\(XY\)[/tex] in [tex]\(\triangle XYZ\)[/tex] corresponds to the side [tex]\(AB\)[/tex] in [tex]\(\triangle ABC\)[/tex]
- The side [tex]\(XZ\)[/tex] in [tex]\(\triangle XYZ\)[/tex] corresponds to the side [tex]\(AC\)[/tex] in [tex]\(\triangle ABC\)[/tex]
Since [tex]\(\triangle XYZ\)[/tex] is a dilation of [tex]\(\triangle ABC\)[/tex] by a scale factor of 5:
- [tex]\(XY = 5 \cdot AB\)[/tex]
- [tex]\(XZ = 5 \cdot AC\)[/tex]
Then:
[tex]\[ \frac{AB}{5 \cdot AB} = \frac{AC}{5 \cdot AC} \][/tex]
Both sides simplify to [tex]\(\frac{1}{5}\)[/tex]. However, this does not directly verify the similarity of [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle XYZ\)[/tex].
Option B: [tex]\(\frac{YZ}{BC} = \frac{AC}{XZ}\)[/tex]
For sides [tex]\(YZ\)[/tex] and [tex]\(BC\)[/tex]:
- The side [tex]\(YZ\)[/tex] in [tex]\(\triangle XYZ\)[/tex] corresponds to the side [tex]\(BC\)[/tex] in [tex]\(\triangle ABC\)[/tex]
- Since [tex]\(\triangle XYZ\)[/tex] is a dilation of [tex]\(\triangle ABC\)[/tex] by a scale factor of 5, [tex]\(YZ = 5 \cdot BC\)[/tex]
Then:
[tex]\[ \frac{5 \cdot BC}{BC} = \frac{AC}{5 \cdot AC} \][/tex]
This gives us:
[tex]\[ 5 = \frac{1}{5} \][/tex]
This is not true, so option B does not verify the similarity.
Option C: [tex]\(\frac{BC}{YZ} = \frac{AB}{XZ}\)[/tex]
Considering [tex]\(BC\)[/tex] and [tex]\(YZ\)[/tex]:
- [tex]\(YZ = 5 \cdot BC\)[/tex]
For sides [tex]\(AB\)[/tex] and [tex]\(XZ\)[/tex]:
- [tex]\(XZ = 5 \cdot AC\)[/tex]
Then:
[tex]\[ \frac{BC}{5 \cdot BC} = \frac{AB}{5 \cdot AC} \][/tex]
Both sides simplify to [tex]\(\frac{1}{5}\)[/tex]. While the ratios match, it does not directly verify similarity in a general manner different from the approach we're analyzing.
Option D: [tex]\(\frac{AB}{AC} = \frac{XZ}{XY}\)[/tex]
Considering the ratios of corresponding sides within each triangle:
- [tex]\(\frac{AB}{AC}\)[/tex] is a ratio of sides in [tex]\(\triangle ABC\)[/tex]
- [tex]\(XZ\)[/tex] and [tex]\(XY\)[/tex] are sides corresponding to [tex]\(AC\)[/tex] and [tex]\(AB\)[/tex] respectively in [tex]\(\triangle XYZ\)[/tex]
Since [tex]\(\triangle XYZ\)[/tex] is a dilation of [tex]\(\triangle ABC\)[/tex] by a scale factor of 5:
- [tex]\(XZ = 5 \cdot AC\)[/tex]
- [tex]\(XY = 5 \cdot AB\)[/tex]
Then:
[tex]\[ \frac{AB}{AC} = \frac{5 \cdot AC}{5 \cdot AB} \][/tex]
Both sides simplify to:
[tex]\[ \frac{AB}{AC} \][/tex]
Thus, the ratios of the corresponding sides are equal, verifying that [tex]\(\triangle ABC\)[/tex] and [tex]\(\triangle XYZ\)[/tex] are similar.
Therefore, the correct option is:
[tex]\[ \boxed{D} \][/tex]
