To analyze the transformations applied to triangle XYZ and determine the relationship between its sides before and after the transformations, let's break down the steps and their effects on the triangle.
1. Translation: The triangle XYZ is translated by the rule [tex]\((x + 1, y - 1)\)[/tex]. This means each point of the triangle is shifted such that 1 is added to the x-coordinate and 1 is subtracted from the y-coordinate.
- If we denote the original coordinates of the vertices of triangle XYZ as [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], and [tex]\((x_3, y_3)\)[/tex], then after the translation, the new coordinates of the vertices become:
- [tex]\((x_1 + 1, y_1 - 1)\)[/tex]
- [tex]\((x_2 + 1, y_2 - 1)\)[/tex]
- [tex]\((x_3 + 1, y_3 - 1)\)[/tex]
- Since translation is a rigid transformation, the shape and size of the triangle do not change. Therefore, the sides of the triangle after translation remain congruent to the sides of the original triangle. This implies that the corresponding sides [tex]\( YZ \)[/tex] and [tex]\( Y'Z' \)[/tex] are congruent after the translation.
2. Dilation: After translation, the triangle is dilated by a scale factor of 4 centered at the origin (0, 0). Dilation enlarges or reduces the size of a figure, while keeping the proportions the same. Since the center is the origin, each coordinate of the triangle is multiplied by 4.
- Translating this to the new vertices, the coordinates after dilation become:
- [tex]\((4(x_1 + 1), 4(y_1 - 1))\)[/tex]
- [tex]\((4(x_2 + 1), 4(y_2 - 1))\)[/tex]
- [tex]\((4(x_3 + 1), 4(y_3 - 1))\)[/tex]
- Since dilation scales all the sides of the triangle by the same factor, the sides of the triangle after dilation are proportional to the sides before the dilation by a factor of 4. This means that the corresponding sides [tex]\( YZ \)[/tex] and [tex]\( Y''Z'' \)[/tex] are proportional after the dilation.
Combining these observations:
- After the translation, the sides of the triangle remain congruent.
- After the dilation, the sides of the triangle are proportional to the sides of the original triangle by a scale factor of 4.
Given these conclusions, the correct statement about the triangles XYZ and X"Y"Z" after the transformations is:
"YZ and Y"Z" are proportional after the dilation and congruent after the translation."
Therefore, the correct answer is:
YZ and Y"Z" are proportional after the dilation and congruent after the translation.
