Understanding the transformations applied to [tex]\( \Delta XYZ \)[/tex] will help us evaluate the properties of [tex]\( \Delta XYZ \)[/tex] and [tex]\( \Delta X'Y'Z' \)[/tex].
1. Reflection over a vertical line:
- This transformation creates a mirror image of [tex]\( \Delta XYZ \)[/tex]. All angles and side lengths remain unchanged, but the orientation can reverse, resulting in a congruent but opposite triangle.
2. Dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex]:
- Dilation reduces the size of [tex]\( \Delta XYZ \)[/tex] such that all side lengths are halved, but the angles remain the same. This results in a similar triangle, [tex]\( \Delta X'Y'Z' \)[/tex], with proportional sides and identical angles to [tex]\( \Delta XYZ \)[/tex].
Given the above transformations, let's analyze each statement:
1. [tex]\( \Delta XYZ \sim \Delta X'Y'Z' \)[/tex]:
- True. A dilation with a scale factor retains the similarity of the triangles. The angles are congruent, and the sides are in proportion (in this case, the sides of [tex]\( \Delta X'Y'Z' \)[/tex] are [tex]\(\frac{1}{2}\)[/tex] the length of the corresponding sides of [tex]\( \Delta XYZ \)[/tex]).
2. [tex]\( \angle XZY \cong \angle YY'X' \)[/tex]:
- This statement appears incorrect because [tex]\( \angle XZY \)[/tex] (an angle in the original triangle) cannot be directly compared to [tex]\( \angle YY'X' \)[/tex], a combination of different vertices not defined in our transformed triangle.
3. [tex]\( \overline{YX} \cong \overline{YX} \)[/tex]:
- This statement is somewhat redundant as it states that a segment is congruent to itself, which is always true, but not informative in the context of changes from the transformations. This is generally a true statement but doesn't provide relevant comparison between two different parts of our transformation process.
4. [tex]\( XZ = 2 X'Z' \)[/tex]:
- True. Since the dilation reduces all sides to [tex]\(\frac{1}{2}\)[/tex], reversing this, the original side lengths would be double those of the dilated triangle. So, [tex]\( XZ\)[/tex] from [tex]\( \Delta XYZ \)[/tex] is indeed [tex]\( 2 \times X'Z' \)[/tex] from [tex]\(\Delta X'Y'Z' \)[/tex].
5. [tex]\( m\angle YXZ = 2 m\angle YXXX \)[/tex]:
- This statement is incorrect. The dilation changes the side lengths but does not alter the angles. Therefore, the measure of angle [tex]\( YXZ \)[/tex] in [tex]\( \Delta XYZ \)[/tex] is equal to the measure of its corresponding angle in [tex]\( \Delta X'Y'Z' \)[/tex].
Thus, the correct options must be:
- [tex]\( \Delta XYZ \sim \Delta X'Y'Z' \)[/tex]
- [tex]\( \overline{YX} \cong \overline{YX} \)[/tex]
- [tex]\( XZ = 2 X'Z' \)[/tex]
