To answer this question, we need to analyze the specific transformations applied to the triangle [tex]\( \triangle XYZ \)[/tex].
1. Reflection over a vertical line:
- Reflections preserve the size and shape of geometric figures (congruence). They also maintain the lengths of sides and measures of angles but change the orientation.
2. Dilation by a scale factor of [tex]\( \frac{1}{2} \)[/tex]:
- Dilation changes the size of the figure but preserves the shape (similarity). With a scale factor of [tex]\( \frac{1}{2} \)[/tex], each side of the triangle will be half its original length, but the measures of the angles will remain the same.
Given the transformations, let's go through each option to determine which statements must be true:
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]:
- Dilation by a scale factor of [tex]\( \frac{1}{2} \)[/tex] will result in a triangle that is similar to the original triangle because the angles remain the same and the sides are proportional (each side of [tex]\( \triangle X'Y'Z' \)[/tex] is half as long as the corresponding side of [tex]\( \triangle XYZ \)[/tex]).
- True
2. [tex]\( \angle XZY \cong \angle Y'Y'X' \)[/tex]:
- For the given angles to be congruent, they must have the same measure. However, [tex]\( \angle XZY \)[/tex] corresponds to [tex]\( \angle X'Z'Y' \)[/tex] (not [tex]\( \angle Y'Y'X' \)[/tex]). Due to the transformations:
- The angle [tex]\( \angle XZY \)[/tex] would correspond to an angle in [tex]\( \triangle X'Y'Z' \)[/tex], but this exact pairing is not [tex]\( \angle Y'Y'X' \)[/tex]. Thus, this statement doesn’t make sense within the context.
- False
3. [tex]\( \overline{YX} \cong \overline{Y'X'} \)[/tex]:
- Reflecting [tex]\( \triangle XYZ \)[/tex] over a vertical line would preserve the lengths of the sides. However, dilating by [tex]\( \frac{1}{2} \)[/tex] would change the length of each side.
- Therefore, [tex]\( \overline{YX} \)[/tex] cannot be congruent to [tex]\( \overline{Y'X'} \)[/tex] because [tex]\( \overline{Y'X'} \)[/tex] is half as long as [tex]\( \overline{YX} \)[/tex].
- False
4. [tex]\( XZ = 2 X'Z' \)[/tex]:
- Since the dilation scale factor is [tex]\( \frac{1}{2} \)[/tex], [tex]\( X'Z' \)[/tex] is half the length of [tex]\( XZ \)[/tex]. This means [tex]\( XZ \)[/tex] is exactly twice the length of [tex]\( X'Z' \)[/tex].
- True
5. [tex]\( m \angle YXZ = 2 m \angle Y'X'Z' \)[/tex]:
- The measures of the angles remain the same during both reflection and dilation. Thus, the measures of the angles do not change. Therefore, [tex]\( m \angle YXZ \)[/tex] cannot be twice [tex]\( m \angle Y'X'Z' \)[/tex] because they are equal.
- False
Therefore, the three true statements about the triangles are:
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]
2. [tex]\( XZ = 2 X'Z' \)[/tex]
3. [tex]\( m \angle YXZ = m \angle Y'X'Z' \)[/tex]
