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Sagot :
To determine which proportion verifies that triangles [tex]\( \triangle ABC \)[/tex] and [tex]\( \triangle XYZ \)[/tex] are similar, we need to use the concept of triangle similarity and the properties of dilation.
When [tex]\( \triangle XYZ \)[/tex] is a dilation of [tex]\( \triangle ABC \)[/tex] by a scale factor of 5, this means that every corresponding side of [tex]\( \triangle XYZ \)[/tex] is exactly 5 times the length of the corresponding side of [tex]\( \triangle ABC \)[/tex]. For similar triangles, the ratios of corresponding sides are equal.
Let us examine the given choices:
Choice A: [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex]
This option suggests that the ratio of side [tex]\( AB \)[/tex] of [tex]\( \triangle ABC \)[/tex] to side [tex]\( XY \)[/tex] of [tex]\( \triangle XYZ \)[/tex] is the same as the ratio of side [tex]\( AC \)[/tex] of [tex]\( \triangle ABC \)[/tex] to side [tex]\( XZ \)[/tex] of [tex]\( \triangle XYZ \)[/tex]. Given that [tex]\( \triangle XYZ \)[/tex] is a dilation of [tex]\( \triangle ABC \)[/tex] by a scale factor of 5, we can express:
[tex]\[ AB \text{ (of } \triangle ABC) \times 5 = XY \text{ (of } \triangle XYZ) \][/tex]
[tex]\[ AC \text{ (of } \triangle ABC) \times 5 = XZ \text{ (of } \triangle XYZ) \][/tex]
Thus, [tex]\(\frac{AB \text{ (of } \triangle ABC)}{XY \text{ (of } \triangle XYZ)} = \frac{1}{5}\)[/tex]
and [tex]\(\frac{AC \text{ (of } \triangle ABC)}{XZ \text{ (of } \triangle XYZ)} = \frac{1}{5}\)[/tex].
So, [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex] holds true and verifies the similarity.
Choice B: [tex]\(\frac{YZ}{BC} = \frac{AC}{XZ}\)[/tex]
For this proportion, we need to analyze:
[tex]\[ YZ \text{ (of } \triangle XYZ) \times 5 = BC \text{ (of } \triangle ABC) \][/tex]
Here’s the relationship for the sides:
[tex]\[ AC \text{ (of } \triangle ABC) \times 5 = XZ \text{ (of } \triangle XYZ) \][/tex]
Similarly,
[tex]\(\frac{YZ \text{ (of } \triangle XYZ)}{BC \text{ (of } \triangle ABC)} = \frac{1}{5}\)[/tex]
and [tex]\(\frac{AC \text{ (of } \triangle ABC)}{XZ \text{ (of } \triangle XYZ)} = \frac{1}{5}\)[/tex]
Even though the same ratio holds, the correct choice must relate side [tex]\( AB \)[/tex] or [tex]\( AC \)[/tex] with [tex]\( XY \)[/tex] or [tex]\( XZ \)[/tex] directly for a specific pair. This relationship does not strictly give us verification since sides [tex]\( YZ \)[/tex] and [tex]\( AC \)[/tex] won't directly enforce verification of their proportional comparative sides.
Choice C: [tex]\(\frac{AB}{AC} = \frac{XZ}{XY}\)[/tex]
This proportion relates the ratios of sides within each triangle to each other:
[tex]\[ \frac{AB}{AC} \][/tex] and [tex]\[ \frac{XZ}{XY} \][/tex]
Given the similarity condition:
[tex]\(\frac{AB \text{ (of } \triangle ABC)}{AC \text{ (of } \triangle ABC)} = \frac{XZ \text{ (of } \triangle XYZ)}{XY \text{ (of } \triangle XYZ)} \)[/tex]
which doesn't align the sides from their respective triangles enforced by scaling/dilation.
Choice D: [tex]\(\frac{BC}{\sqrt{7}} = \frac{AB}{\sqrt{7}}\)[/tex]
Here, the ratios involve side [tex]\( BC \)[/tex] and sides not inherently aligned between triangles, thus not aiding similarity verification.
Nothing matches intended proportional criteria revealed by side comparison directly linked to scaling. The valid verification stands clear with
[tex]\[\boxed{\text{A}}\][/tex]
The proportion [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex] correctly verifies that [tex]\( \triangle ABC \)[/tex] and [tex]\( \triangle XYZ \)[/tex] are similar considering the applied scale factor 5.
When [tex]\( \triangle XYZ \)[/tex] is a dilation of [tex]\( \triangle ABC \)[/tex] by a scale factor of 5, this means that every corresponding side of [tex]\( \triangle XYZ \)[/tex] is exactly 5 times the length of the corresponding side of [tex]\( \triangle ABC \)[/tex]. For similar triangles, the ratios of corresponding sides are equal.
Let us examine the given choices:
Choice A: [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex]
This option suggests that the ratio of side [tex]\( AB \)[/tex] of [tex]\( \triangle ABC \)[/tex] to side [tex]\( XY \)[/tex] of [tex]\( \triangle XYZ \)[/tex] is the same as the ratio of side [tex]\( AC \)[/tex] of [tex]\( \triangle ABC \)[/tex] to side [tex]\( XZ \)[/tex] of [tex]\( \triangle XYZ \)[/tex]. Given that [tex]\( \triangle XYZ \)[/tex] is a dilation of [tex]\( \triangle ABC \)[/tex] by a scale factor of 5, we can express:
[tex]\[ AB \text{ (of } \triangle ABC) \times 5 = XY \text{ (of } \triangle XYZ) \][/tex]
[tex]\[ AC \text{ (of } \triangle ABC) \times 5 = XZ \text{ (of } \triangle XYZ) \][/tex]
Thus, [tex]\(\frac{AB \text{ (of } \triangle ABC)}{XY \text{ (of } \triangle XYZ)} = \frac{1}{5}\)[/tex]
and [tex]\(\frac{AC \text{ (of } \triangle ABC)}{XZ \text{ (of } \triangle XYZ)} = \frac{1}{5}\)[/tex].
So, [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex] holds true and verifies the similarity.
Choice B: [tex]\(\frac{YZ}{BC} = \frac{AC}{XZ}\)[/tex]
For this proportion, we need to analyze:
[tex]\[ YZ \text{ (of } \triangle XYZ) \times 5 = BC \text{ (of } \triangle ABC) \][/tex]
Here’s the relationship for the sides:
[tex]\[ AC \text{ (of } \triangle ABC) \times 5 = XZ \text{ (of } \triangle XYZ) \][/tex]
Similarly,
[tex]\(\frac{YZ \text{ (of } \triangle XYZ)}{BC \text{ (of } \triangle ABC)} = \frac{1}{5}\)[/tex]
and [tex]\(\frac{AC \text{ (of } \triangle ABC)}{XZ \text{ (of } \triangle XYZ)} = \frac{1}{5}\)[/tex]
Even though the same ratio holds, the correct choice must relate side [tex]\( AB \)[/tex] or [tex]\( AC \)[/tex] with [tex]\( XY \)[/tex] or [tex]\( XZ \)[/tex] directly for a specific pair. This relationship does not strictly give us verification since sides [tex]\( YZ \)[/tex] and [tex]\( AC \)[/tex] won't directly enforce verification of their proportional comparative sides.
Choice C: [tex]\(\frac{AB}{AC} = \frac{XZ}{XY}\)[/tex]
This proportion relates the ratios of sides within each triangle to each other:
[tex]\[ \frac{AB}{AC} \][/tex] and [tex]\[ \frac{XZ}{XY} \][/tex]
Given the similarity condition:
[tex]\(\frac{AB \text{ (of } \triangle ABC)}{AC \text{ (of } \triangle ABC)} = \frac{XZ \text{ (of } \triangle XYZ)}{XY \text{ (of } \triangle XYZ)} \)[/tex]
which doesn't align the sides from their respective triangles enforced by scaling/dilation.
Choice D: [tex]\(\frac{BC}{\sqrt{7}} = \frac{AB}{\sqrt{7}}\)[/tex]
Here, the ratios involve side [tex]\( BC \)[/tex] and sides not inherently aligned between triangles, thus not aiding similarity verification.
Nothing matches intended proportional criteria revealed by side comparison directly linked to scaling. The valid verification stands clear with
[tex]\[\boxed{\text{A}}\][/tex]
The proportion [tex]\(\frac{AB}{XY} = \frac{AC}{XZ}\)[/tex] correctly verifies that [tex]\( \triangle ABC \)[/tex] and [tex]\( \triangle XYZ \)[/tex] are similar considering the applied scale factor 5.
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