To determine which statements about triangles [tex]$\triangle X Y Z$[/tex] and [tex]$\Delta X^{\prime} Y^{\prime} Z^{\prime}$[/tex] must be true after [tex]$\triangle X Y Z$[/tex] is reflected over a vertical line and then dilated by a scale factor of [tex]\(\frac{1}{2}\)[/tex], let's analyze each statement one by one.
1. [tex]$\triangle X Y Z \sim \triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex]:
- When a triangle undergoes a reflection followed by a dilation, the resulting triangle retains the same shape but may differ in size. Thus, the triangles remain similar (all corresponding angles are equal, and the sides are proportional). Therefore, [tex]$\triangle X Y Z \sim \triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex].
2. [tex]$\angle X Z Y=\angle Y^{\prime} Z^{\prime} X^{\prime}$[/tex]:
- One of the properties of similar triangles is that corresponding angles are equal. Therefore, the corresponding angles of the original and the transformed triangles are equal. This implies [tex]$\angle X Z Y = \angle Y^{\prime} Z^{\prime} X^{\prime}$[/tex].
3. [tex]$\overline{Y X}=\overline{Y^{\prime} X^{\prime}}$[/tex]:
- After the triangle is reflected and then dilated by a scale factor of [tex]\(\frac{1}{2}\)[/tex], the side lengths change proportionally. The statement [tex]$\overline{Y X} = \overline{Y^{\prime} X^{\prime}}$[/tex] would only hold true if no scaling occurred, but since we have a dilation by [tex]\(\frac{1}{2}\)[/tex], the side lengths of [tex]$\Delta X^{\prime} Y^{\prime} Z^{\prime}$[/tex] are actually half those of [tex]$\triangle X Y Z$[/tex].
4. [tex]$X Z=2 X^{\prime} Z^{\prime}$[/tex]:
- Given the scale factor of [tex]\(\frac{1}{2}\)[/tex], each side length of the original triangle is twice as long as the corresponding side length in the transformed triangle. Thus, [tex]$X Z = 2 \cdot X^{\prime} Z^{\prime}$[/tex] is true.
5. [tex]$m \angle Y X Z=2 m \angle Y^{\prime} X^{\prime} Z^{\prime}$[/tex]:
- When triangles are similar, corresponding angles remain equal, not scaled. The multiplication of the measure of angles by a factor (such as 2) is incorrect. Therefore, [tex]$m \angle Y X Z=2 m \angle Y^{\prime} X^{\prime} Z^{\prime}$[/tex] is not true.
Considering the above analysis, the correct options are:
1. [tex]$\triangle X Y Z \sim \triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex].
2. [tex]$\angle X Z Y=\angle Y^{\prime} Z^{\prime} X^{\prime}$[/tex].
4. [tex]$X Z=2 X^{\prime} Z^{\prime}$[/tex].
