Which series of transformations shows that rectangle W'X'Y'Z' is similar to rectangle WXYZ? Assume all dilations are centered at the origin.
A.
Reflect rectangle XYZW across side WZ, translate it seven units up and two units to the left, and dilate it by a scale factor of 3.
B.
Reflect rectangle XYZW across side WZ, translate it two units up and seven units to the left, and dilate it by a scale factor of .
C.
Reflect rectangle XYZW across side WX, translate it five units up and two units to the left, and dilate it by a scale factor of .
D.
Reflect rectangle XYZW across side WX, translate it two units up and five units to the left, and dilate it by a scale factor of 3.
The correct answer is D. Reflecting the rectangle across side WX and then translating it two units up and five units to the left is equivalent to reflecting it across the origin and then translating it five units up and two units to the left. Finally, dilating it by a scale factor of 3 centered at the origin will result in the similarity between rectangle W'X'Y'Z' and rectangle WXYZ.
The correct series of transformations that shows that rectangle W'X'Y'Z' is similar to rectangle WXYZ is option D.
In option D, rectangle XYZW is reflected across side WX, then translated two units up and five units to the left, and finally dilated by a scale factor of 3.
This series of transformations preserves both shape and size, which is the definition of similarity. Therefore, option D is the correct answer.
To determine which series of transformations shows that rectangle W'X'Y'Z' is similar to rectangle WXYZ, we need to compare the given options to the definition of similar figures.
Similar figures have the following properties:
1. Corresponding angles are congruent.
2. Corresponding sides are proportional.
Let's analyze each option:
A. Reflect rectangle XYZW across side WZ, translate it seven units up and two units to the left, and dilate it by a scale factor of 3.
This option reflects the rectangle across side WZ, translates it, and then dilates it. However, it doesn't mention the scale factor used for dilation. So, we cannot determine if the corresponding sides are proportional. Therefore, we can eliminate option A.
B. Reflect rectangle XYZW across side WZ, translate it two units up and seven units to the left, and dilate it by a scale factor of .
This option reflects the rectangle across side WZ, translates it, and dilates it by an unknown scale factor. Without knowing the scale factor, we cannot verify if the corresponding sides are proportional. Therefore, we can eliminate option B.
C. Reflect rectangle XYZW across side WX, translate it five units up and two units to the left, and dilate it by a scale factor of .
This option reflects the rectangle across side WX, translates it, and dilates it by an unknown scale factor. Without knowing the scale factor, we cannot determine if the corresponding sides are proportional. Therefore, we can eliminate option C.
D. Reflect rectangle XYZW across side WX, translate it two units up and five units to the left, and dilate it by a scale factor of 3.
This option reflects the rectangle across side WX, translates it, and dilates it by a scale factor of 3. Since the scale factor is specified, we can determine that the corresponding sides are proportional, as required for similar figures. Therefore, option D is the correct series of transformations showing that rectangle W'X'Y'Z' is similar to rectangle WXYZ.