Certainly! Let's analyze the transformations applied to triangle \( \triangle XYZ \):
1. Reflection over a vertical line:
- Reflecting \( \triangle XYZ \) over a vertical line means that each point of the triangle is mirrored to a corresponding point on the opposite side of the reflection line. This transformation preserves the sizes and shapes of the triangle's angles and sides, but changes their orientation.
2. Dilation by a scale factor of \( \frac{1}{2} \):
- After reflection, a dilation with a scale factor of \( \frac{1}{2} \) reduces the size of the triangle to half of its original dimensions, maintaining the shape and the proportional relationships between the sides and angles.
Considering these transformations, we can determine the truths about the triangles \( \triangle XYZ \) and \( \triangle X'Y'Z' \):
1. Similarity \( \triangle XYZ \sim \triangle X'Y'Z' \):
- As the dilation transformation retains the shape and the reflective transformation preserves the angles, \( \triangle XYZ \) and \( \triangle X'Y'Z' \) must be similar triangles.
2. Angle preservation \( \angle XZY = \angle Y'Z'X' \):
- Reflecting and dilating the triangle does not alter the measure of any internal angles of the triangle; therefore, corresponding angles in \( \triangle XYZ \) and \( \triangle X'Y'Z' \) are equal.
3. Side length relation \( XZ = 2 \cdot X'Z' \):
- The dilation transformation reduces the original side lengths by half. Hence, for corresponding sides, \( XZ \) in \( \triangle XYZ \) will be twice the length of \( X'Z' \) in \( \triangle X'Y'Z' \).
To conclude:
- \( \triangle XYZ \sim \triangle X'Y'Z' \)
- \( \angle XZY = \angle Y'Z'X' \)
- \( XZ = 2 X'Z' \)
These options are correct:
- \( \triangle XYZ \sim \triangle X'Y'Z' \)
- \( \angle X Z Y = \angle Y' Z' X' \)
- [tex]\( X Z = 2 X' Z' \)[/tex]
