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Answer :
To determine whether pool STUV is similar to pool WXYZ, you can use geometric transformation techniques. Here's the detailed explanation:
1. Translation: First, translate the entire pool WXYZ so that point W of WXYZ coincides with point S of STUV. This means you are shifting the whole pool WXYZ in the plane so that point W aligns exactly with point S. This translation maintains the shapes and size of the original figure WXYZ.
2. Dilation: After the translation, you need to dilate (resize) pool WXYZ. The dilation factor is determined by the ratio of the sides TS and XW. Specifically, the dilation should be by the ratio [tex]\( \frac{\overline{TS}}{\overline{XW}} \)[/tex].
- Why the ratio [tex]\( \frac{\overline{TS}}{\overline{XW}} \)[/tex]? Dilation by this specific ratio ensures that every corresponding side of WXYZ is scaled proportionately to match the corresponding sides of STUV, thus maintaining the similarity between the shapes.
3. Check for Similarity: After the translation and dilation, pool WXYZ will have the same shape as STUV if they are similar. Since similarity in geometry means that corresponding angles are equal and the lengths of corresponding sides are proportional, performing these steps ensures that these criteria are met.
Thus, the correct statement that explains how the company can determine whether pool STUV is similar to pool WXYZ is:
"Translate WXYZ so that point W of WXYZ lies on point S of STUV, then dilate WXYZ by the ratio [tex]\( \frac{\overline{TS}}{\overline{XW}} \)[/tex]."
1. Translation: First, translate the entire pool WXYZ so that point W of WXYZ coincides with point S of STUV. This means you are shifting the whole pool WXYZ in the plane so that point W aligns exactly with point S. This translation maintains the shapes and size of the original figure WXYZ.
2. Dilation: After the translation, you need to dilate (resize) pool WXYZ. The dilation factor is determined by the ratio of the sides TS and XW. Specifically, the dilation should be by the ratio [tex]\( \frac{\overline{TS}}{\overline{XW}} \)[/tex].
- Why the ratio [tex]\( \frac{\overline{TS}}{\overline{XW}} \)[/tex]? Dilation by this specific ratio ensures that every corresponding side of WXYZ is scaled proportionately to match the corresponding sides of STUV, thus maintaining the similarity between the shapes.
3. Check for Similarity: After the translation and dilation, pool WXYZ will have the same shape as STUV if they are similar. Since similarity in geometry means that corresponding angles are equal and the lengths of corresponding sides are proportional, performing these steps ensures that these criteria are met.
Thus, the correct statement that explains how the company can determine whether pool STUV is similar to pool WXYZ is:
"Translate WXYZ so that point W of WXYZ lies on point S of STUV, then dilate WXYZ by the ratio [tex]\( \frac{\overline{TS}}{\overline{XW}} \)[/tex]."
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