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Answer :
To determine how the pool shapes STUV and WXYZ are similar, we need to use the concepts of translation and dilation in geometry.
1. Translation: This is a type of transformation that slides a shape in the plane without changing its size, shape, or orientation. For two shapes to be similar through translation alone, one shape must be positioned over the other using this sliding motion.
2. Dilation: This is another transformation that alters the size of a shape but maintains its proportions. The shape is either enlarged or reduced by a certain ratio, known as the scale factor.
For two shapes to be similar, we need to position one shape over the other and possibly scale it.
### Step-by-step Explanation:
- Translate Point X of WXYZ to Point T of STUV: Start by moving shape WXYZ such that its point X is directly over point T of STUV. This translation places part of the shape in alignment with the second shape, but it won't yet confirm similarity unless both shapes now have corresponding sides and angles that are proportionally equal.
- Dilation by Ratio of Segments: After translating, you would dilate shape WXYZ using a scale factor. The scale factor should be the ratio of the lengths of corresponding segments from the two shapes. If XW and TS are corresponding segments from WXYZ and STUV respectively, then the scale factor is given by the ratio of these two segments, [tex]\(\frac{\overline{XW}}{\overline{TS}}\)[/tex].
This series of transformations confirms that shape WXYZ is resized to match STUV proportionally, validating their similarity.
In conclusion, the correct statement describes translating WXYZ to align point X over point T, followed by dilating the shape using the appropriate scale factor, [tex]\(\frac{\overline{XW}}{\overline{TS}}\)[/tex].
1. Translation: This is a type of transformation that slides a shape in the plane without changing its size, shape, or orientation. For two shapes to be similar through translation alone, one shape must be positioned over the other using this sliding motion.
2. Dilation: This is another transformation that alters the size of a shape but maintains its proportions. The shape is either enlarged or reduced by a certain ratio, known as the scale factor.
For two shapes to be similar, we need to position one shape over the other and possibly scale it.
### Step-by-step Explanation:
- Translate Point X of WXYZ to Point T of STUV: Start by moving shape WXYZ such that its point X is directly over point T of STUV. This translation places part of the shape in alignment with the second shape, but it won't yet confirm similarity unless both shapes now have corresponding sides and angles that are proportionally equal.
- Dilation by Ratio of Segments: After translating, you would dilate shape WXYZ using a scale factor. The scale factor should be the ratio of the lengths of corresponding segments from the two shapes. If XW and TS are corresponding segments from WXYZ and STUV respectively, then the scale factor is given by the ratio of these two segments, [tex]\(\frac{\overline{XW}}{\overline{TS}}\)[/tex].
This series of transformations confirms that shape WXYZ is resized to match STUV proportionally, validating their similarity.
In conclusion, the correct statement describes translating WXYZ to align point X over point T, followed by dilating the shape using the appropriate scale factor, [tex]\(\frac{\overline{XW}}{\overline{TS}}\)[/tex].
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