Question

Triangle XYZ is translated by the rule (x 1, y - 1) and then dilated by a scale factor of 4 centered at the origin. Which statement describes the properties of triangles XYZ and X"Y"Z" after the transformations?

1) Y and ∠Y" are congruent after the translation, but not after the dilation.
2) ∠Y and ∠Y" are congruent after the dilation, but not after the translation.
3) Segment YZ and segment Y"Z" are proportional after the translation, but not after the dilation.
4) Segment YZ and segment Y"Z" are proportional after the dilation and congruent after the translation.

Answer 1

Triangles XYZ and X"Y"Z" are congruent after the translation and angles remain congruent after both the translation and dilation. The length of sides becomes proportional after the subsequent dilation by a factor of 4 centered at the origin.

Step-by-step explanation:

The properties of triangles XYZ and X"Y"Z" after the transformations can be described as follows:

1. After the translation by the rule (x + 1, y - 1), the shapes of the triangles remain unchanged, meaning angles and side lengths are preserved, therefore, ∆XYZ and ∆X"Y"Z" are congruent after the translation. The corresponding sides and angles remain equal, which includes ∠Y and ∠Y".
2. The subsequent dilation by a factor of 4 centered at the origin will change the size of the triangle but not the shape. This means that ∠Y and ∠Y" remain congruent after the dilation; however, the sides of the triangle are multiplied by 4, making segments XYZ and X"Y"Z" proportional but not congruent after the dilation.

Therefore, the correct statement that describes the properties of the triangles after the transformations is: ∠Y and ∠Y" are congruent after the dilation, but not after the translation, and Segment YZ and segment Y"Z" are proportional after the dilation and congruent after the translation.