Answer :
Certainly! Let's carefully consider the transformations and their implications on the properties of the triangles involved.
1. Reflection Over a Vertical Line:
- Reflecting a triangle over a vertical line does not change the size of the angles, it only changes the orientation. Therefore, the reflected triangle is congruent to the original triangle in terms of angles and side lengths, but its position and/or orientation is mirrored.
2. Dilation by a Scale Factor of [tex]\( \frac{1}{2} \)[/tex]:
- Dilation by a scale factor of [tex]\( \frac{1}{2} \)[/tex] shrinks the triangle to half its original size while preserving the proportional relationships between the side lengths and the measures of the angles.
Given these transformations, let's determine what must be true about [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle XYY\)[/tex] after reflection and dilation:
### 1. Angle Preservation
Since dilation preserves the angles of the triangle but scales the side lengths by a factor of [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ \angle XZY = \angle YZX \][/tex]
This tells us that the angles in [tex]\(\triangle XYZ\)[/tex] (before transformation) are the same as the corresponding angles in [tex]\(\triangle XYY\)[/tex] (after transformation). Hence:
[tex]\(\angle XZY \approx \angle YZX\)[/tex] is true.
### 2. Side Length Preservation (Proportionality)
Considering the reflection does not change segment lengths, and dilation halves the lengths of all sides:
[tex]\[ \overline{YX} = \overline{YX} \][/tex]
Even after reflection and dilation, the segment [tex]\(YX\)[/tex] in [tex]\(\triangle XYY\)[/tex] is directly half of the segment [tex]\(YX\)[/tex] in [tex]\(\triangle XYZ\)[/tex]. Therefore, side lengths are proportional, confirming:
[tex]\(\overline{YX} = \overline{YX}\)[/tex] is true.
### 3. Side Length Relationship Due to Dilation
Since the dilation reduced all sides by a factor of [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ XZ = 2 \cdot XZ \][/tex]
After dilation, any segment in [tex]\(\triangle XYY\)[/tex] should be half the length of the corresponding segment in [tex]\(\triangle XYZ\)[/tex], so:
[tex]\(\overline{XZ} = 2 \cdot \overline{XZ}\)[/tex] is also true.
### Conclusion
Based on the transformations (reflection and dilation), we can confirm the following statements must be true:
1. [tex]\(\angle XZY \approx \angle YZX\)[/tex]
2. [tex]\(\overline{YX} = \overline{YX}\)[/tex]
3. [tex]\(\overline{XZ} = 2 \cdot \overline{XZ}\)[/tex]
Hence, the correct options are:
- [tex]\(\angle XZY \approx \angle YZX\)[/tex]
- [tex]\(\overline{YX} = \overline{YX}\)[/tex]
- [tex]\(\overline{XZ} = 2 \overline{XZ}\)[/tex]
1. Reflection Over a Vertical Line:
- Reflecting a triangle over a vertical line does not change the size of the angles, it only changes the orientation. Therefore, the reflected triangle is congruent to the original triangle in terms of angles and side lengths, but its position and/or orientation is mirrored.
2. Dilation by a Scale Factor of [tex]\( \frac{1}{2} \)[/tex]:
- Dilation by a scale factor of [tex]\( \frac{1}{2} \)[/tex] shrinks the triangle to half its original size while preserving the proportional relationships between the side lengths and the measures of the angles.
Given these transformations, let's determine what must be true about [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle XYY\)[/tex] after reflection and dilation:
### 1. Angle Preservation
Since dilation preserves the angles of the triangle but scales the side lengths by a factor of [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ \angle XZY = \angle YZX \][/tex]
This tells us that the angles in [tex]\(\triangle XYZ\)[/tex] (before transformation) are the same as the corresponding angles in [tex]\(\triangle XYY\)[/tex] (after transformation). Hence:
[tex]\(\angle XZY \approx \angle YZX\)[/tex] is true.
### 2. Side Length Preservation (Proportionality)
Considering the reflection does not change segment lengths, and dilation halves the lengths of all sides:
[tex]\[ \overline{YX} = \overline{YX} \][/tex]
Even after reflection and dilation, the segment [tex]\(YX\)[/tex] in [tex]\(\triangle XYY\)[/tex] is directly half of the segment [tex]\(YX\)[/tex] in [tex]\(\triangle XYZ\)[/tex]. Therefore, side lengths are proportional, confirming:
[tex]\(\overline{YX} = \overline{YX}\)[/tex] is true.
### 3. Side Length Relationship Due to Dilation
Since the dilation reduced all sides by a factor of [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ XZ = 2 \cdot XZ \][/tex]
After dilation, any segment in [tex]\(\triangle XYY\)[/tex] should be half the length of the corresponding segment in [tex]\(\triangle XYZ\)[/tex], so:
[tex]\(\overline{XZ} = 2 \cdot \overline{XZ}\)[/tex] is also true.
### Conclusion
Based on the transformations (reflection and dilation), we can confirm the following statements must be true:
1. [tex]\(\angle XZY \approx \angle YZX\)[/tex]
2. [tex]\(\overline{YX} = \overline{YX}\)[/tex]
3. [tex]\(\overline{XZ} = 2 \cdot \overline{XZ}\)[/tex]
Hence, the correct options are:
- [tex]\(\angle XZY \approx \angle YZX\)[/tex]
- [tex]\(\overline{YX} = \overline{YX}\)[/tex]
- [tex]\(\overline{XZ} = 2 \overline{XZ}\)[/tex]
