Answer :
To determine which proportions verify that [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle UVW\)[/tex] are similar under a scale factor of 1, we need to understand the properties of similar triangles. When two triangles are similar, their corresponding sides are proportional, and their corresponding angles are congruent.
Given that the scale factor is 1, the triangles are identical in size, making all corresponding sides equal. Still, proportional relationships must be maintained.
Let's analyze each given option to find the correct proportion:
### Option A: [tex]\(\frac{RS}{UV} = \frac{UW}{SR}\)[/tex]
To check this proportion, we need to see if it maintains the correspondence between the sides of the two triangles.
- [tex]\(\frac{RS}{UV}\)[/tex]: This ratio compares the side [tex]\(RS\)[/tex] of [tex]\(\triangle RST\)[/tex] with the corresponding side [tex]\(UV\)[/tex] of [tex]\(\triangle UVW\)[/tex]. This part is correct since [tex]\(RS\)[/tex] corresponds to [tex]\(UV\)[/tex].
- [tex]\(\frac{UW}{SR}\)[/tex]: This ratio compares side [tex]\(UW\)[/tex] of [tex]\(\triangle UVW\)[/tex] with side [tex]\(SR\)[/tex] of [tex]\(\triangle RST\)[/tex]. This part is not maintaining the corresponding side relationship because [tex]\(SR\)[/tex] does not correspond to [tex]\(UW\)[/tex].
Thus, option A does not correctly verify the similarity between the triangles.
### Option B: [tex]\(\frac{RS}{UV} = \frac{RT}{UW}\)[/tex]
Let's check this proportion:
- [tex]\(\frac{RS}{UV}\)[/tex]: This ratio compares the side [tex]\(RS\)[/tex] of [tex]\(\triangle RST\)[/tex] with the corresponding side [tex]\(UV\)[/tex] of [tex]\(\triangle UVW\)[/tex], which is correct.
- [tex]\(\frac{RT}{UW}\)[/tex]: This ratio compares the side [tex]\(RT\)[/tex] of [tex]\(\triangle RST\)[/tex] with the corresponding side [tex]\(UW\)[/tex] of [tex]\(\triangle UVW\)[/tex], which is also correct.
Since both parts correctly compare the corresponding sides of the triangles, option B verifies the similarity between [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle UVW\)[/tex].
### Option C: [tex]\(\frac{RS}{ST} = \frac{W}{VU}\)[/tex]
Let's check this proportion:
- [tex]\(\frac{RS}{ST}\)[/tex]: This ratio correctly compares two sides ([tex]\(RS\)[/tex] and [tex]\(ST\)[/tex]) of [tex]\(\triangle RST\)[/tex].
- [tex]\(\frac{W}{VU}\)[/tex]: This ratio, however, does not correctly represent a side ratio of [tex]\(\triangle UVW\)[/tex] (specifically, [tex]\(W\)[/tex] is not a side of the triangle in the same sense as the other ratios), so it is incorrect.
Thus, option C does not correctly verify the similarity.
### Option D: [tex]\(\frac{RT}{ST} = \frac{UW}{UV}\)[/tex]
Let's check this proportion:
- [tex]\(\frac{RT}{ST}\)[/tex]: This ratio correctly compares two sides ([tex]\(RT\)[/tex] and [tex]\(ST\)[/tex]) of [tex]\(\triangle RST\)[/tex].
- [tex]\(\frac{UW}{UV}\)[/tex]: This ratio correctly compares two sides ([tex]\(UW\)[/tex] and [tex]\(UV\)[/tex]) of [tex]\(\triangle UVW\)[/tex].
At first glance, this might seem correct, but these ratios do not directly confirm the similarity between the triangles as they involve two different ratios within a single triangle rather than comparing corresponding sides between the two triangles.
The only option that accurately verifies the similarity by comparing corresponding sides is:
[tex]\[ \boxed{\text{B: } \frac{RS}{UV} = \frac{RT}{UW}} \][/tex]
Given that the scale factor is 1, the triangles are identical in size, making all corresponding sides equal. Still, proportional relationships must be maintained.
Let's analyze each given option to find the correct proportion:
### Option A: [tex]\(\frac{RS}{UV} = \frac{UW}{SR}\)[/tex]
To check this proportion, we need to see if it maintains the correspondence between the sides of the two triangles.
- [tex]\(\frac{RS}{UV}\)[/tex]: This ratio compares the side [tex]\(RS\)[/tex] of [tex]\(\triangle RST\)[/tex] with the corresponding side [tex]\(UV\)[/tex] of [tex]\(\triangle UVW\)[/tex]. This part is correct since [tex]\(RS\)[/tex] corresponds to [tex]\(UV\)[/tex].
- [tex]\(\frac{UW}{SR}\)[/tex]: This ratio compares side [tex]\(UW\)[/tex] of [tex]\(\triangle UVW\)[/tex] with side [tex]\(SR\)[/tex] of [tex]\(\triangle RST\)[/tex]. This part is not maintaining the corresponding side relationship because [tex]\(SR\)[/tex] does not correspond to [tex]\(UW\)[/tex].
Thus, option A does not correctly verify the similarity between the triangles.
### Option B: [tex]\(\frac{RS}{UV} = \frac{RT}{UW}\)[/tex]
Let's check this proportion:
- [tex]\(\frac{RS}{UV}\)[/tex]: This ratio compares the side [tex]\(RS\)[/tex] of [tex]\(\triangle RST\)[/tex] with the corresponding side [tex]\(UV\)[/tex] of [tex]\(\triangle UVW\)[/tex], which is correct.
- [tex]\(\frac{RT}{UW}\)[/tex]: This ratio compares the side [tex]\(RT\)[/tex] of [tex]\(\triangle RST\)[/tex] with the corresponding side [tex]\(UW\)[/tex] of [tex]\(\triangle UVW\)[/tex], which is also correct.
Since both parts correctly compare the corresponding sides of the triangles, option B verifies the similarity between [tex]\(\triangle RST\)[/tex] and [tex]\(\triangle UVW\)[/tex].
### Option C: [tex]\(\frac{RS}{ST} = \frac{W}{VU}\)[/tex]
Let's check this proportion:
- [tex]\(\frac{RS}{ST}\)[/tex]: This ratio correctly compares two sides ([tex]\(RS\)[/tex] and [tex]\(ST\)[/tex]) of [tex]\(\triangle RST\)[/tex].
- [tex]\(\frac{W}{VU}\)[/tex]: This ratio, however, does not correctly represent a side ratio of [tex]\(\triangle UVW\)[/tex] (specifically, [tex]\(W\)[/tex] is not a side of the triangle in the same sense as the other ratios), so it is incorrect.
Thus, option C does not correctly verify the similarity.
### Option D: [tex]\(\frac{RT}{ST} = \frac{UW}{UV}\)[/tex]
Let's check this proportion:
- [tex]\(\frac{RT}{ST}\)[/tex]: This ratio correctly compares two sides ([tex]\(RT\)[/tex] and [tex]\(ST\)[/tex]) of [tex]\(\triangle RST\)[/tex].
- [tex]\(\frac{UW}{UV}\)[/tex]: This ratio correctly compares two sides ([tex]\(UW\)[/tex] and [tex]\(UV\)[/tex]) of [tex]\(\triangle UVW\)[/tex].
At first glance, this might seem correct, but these ratios do not directly confirm the similarity between the triangles as they involve two different ratios within a single triangle rather than comparing corresponding sides between the two triangles.
The only option that accurately verifies the similarity by comparing corresponding sides is:
[tex]\[ \boxed{\text{B: } \frac{RS}{UV} = \frac{RT}{UW}} \][/tex]
