Answer :
Let's analyze the transformations applied to [tex]\(\triangle XYZ\)[/tex] and determine what must be true for the properties of [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle X'Y'Z'\)[/tex].
### 1. Reflection Over a Vertical Line
Reflecting [tex]\(\triangle XYZ\)[/tex] over a vertical line means that the triangle is flipped horizontally. This transformation preserves the size and shape of the triangle, meaning that:
- The angles remain the same.
- The side lengths remain the same (though their positions are mirrored).
### 2. Dilation by a Scale Factor of [tex]\(\frac{1}{2}\)[/tex]
Dilating the triangle by a scale factor of [tex]\(\frac{1}{2}\)[/tex] reduces the size of the triangle by half. This transformation affects the triangles in the following way:
- The corresponding angles remain equal.
- The side lengths are reduced to half of their original lengths.
Given these transformations, let's examine the given options to determine which are true:
#### Option 1: [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
Similarity between triangles ([tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]) means that they have the same shape but not necessarily the same size. This similarity condition is met if they have:
- Corresponding angles that are equal.
- Corresponding sides that are proportional.
Since [tex]\(\triangle XYZ\)[/tex] is reduced by a scale factor of [tex]\(\frac{1}{2}\)[/tex], the triangles are indeed similar:
[tex]\[ \triangle XYZ \sim \triangle X'Y'Z' \][/tex]
#### Option 2: [tex]\(\angle XZY \cong \angle YZX\)[/tex]
This option compares specific angles within the same triangle [tex]\(\triangle XYZ\)[/tex]. Note that reflecting and dilating do not change the internal angle relationships within a single triangle. Based on the given transformations, this does not provide any relevant comparison between [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle X'Y'Z'\)[/tex]. Therefore, it is not necessarily true based on the transformations.
#### Option 3: [tex]\(\overline{YX} \cong \overline{Y'X}\)[/tex]
Congruency between [tex]\(\overline{YX}\)[/tex] and [tex]\(\overline{Y'X'}\)[/tex] would imply that the lengths of these segments are identical. However, due to the dilation by a factor of [tex]\(\frac{1}{2}\)[/tex], the corresponding segments in [tex]\(\triangle X'Y'Z'\)[/tex] are half the length of the original segments in [tex]\(\triangle XYZ\)[/tex]. Therefore, this statement is false.
#### Option 4: [tex]\(XZ = 2X'Z'\)[/tex]
Since the dilation reduces all side lengths by a factor of [tex]\(\frac{1}{2}\)[/tex], the new length [tex]\(X'Z'\)[/tex] is [tex]\(\frac{1}{2}\)[/tex] the length of [tex]\(XZ\)[/tex]. Hence,
[tex]\[ XZ = 2X'Z' \][/tex]
This statement is true.
#### Option 5: [tex]\(m \angle YXZ = 2 m \angle Y'X'Z'\)[/tex]
This statement suggests that the measure of angle [tex]\(YXZ\)[/tex] is twice the measure of angle [tex]\(Y'X'Z'\)[/tex]. Angles are preserved during reflections and dilations, which means that the measures of the corresponding angles remain the same. Therefore, this statement is false.
### Conclusion
Based on our analysis, the three statements that must be true are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(\angle XZY \cong \angle YZX\)[/tex]
3. [tex]\(XZ = 2X'Z'\)[/tex]
Thus, the correct choices are:
- [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
- [tex]\(\overline{XZ} = 2 \overline{X'Z'}\)[/tex]
- [tex]\(\angle XZY \cong \angle YZX\)[/tex]
Hence, the final answer is:
[tex]\[ (1, 2, 4) \][/tex]
### 1. Reflection Over a Vertical Line
Reflecting [tex]\(\triangle XYZ\)[/tex] over a vertical line means that the triangle is flipped horizontally. This transformation preserves the size and shape of the triangle, meaning that:
- The angles remain the same.
- The side lengths remain the same (though their positions are mirrored).
### 2. Dilation by a Scale Factor of [tex]\(\frac{1}{2}\)[/tex]
Dilating the triangle by a scale factor of [tex]\(\frac{1}{2}\)[/tex] reduces the size of the triangle by half. This transformation affects the triangles in the following way:
- The corresponding angles remain equal.
- The side lengths are reduced to half of their original lengths.
Given these transformations, let's examine the given options to determine which are true:
#### Option 1: [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
Similarity between triangles ([tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]) means that they have the same shape but not necessarily the same size. This similarity condition is met if they have:
- Corresponding angles that are equal.
- Corresponding sides that are proportional.
Since [tex]\(\triangle XYZ\)[/tex] is reduced by a scale factor of [tex]\(\frac{1}{2}\)[/tex], the triangles are indeed similar:
[tex]\[ \triangle XYZ \sim \triangle X'Y'Z' \][/tex]
#### Option 2: [tex]\(\angle XZY \cong \angle YZX\)[/tex]
This option compares specific angles within the same triangle [tex]\(\triangle XYZ\)[/tex]. Note that reflecting and dilating do not change the internal angle relationships within a single triangle. Based on the given transformations, this does not provide any relevant comparison between [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle X'Y'Z'\)[/tex]. Therefore, it is not necessarily true based on the transformations.
#### Option 3: [tex]\(\overline{YX} \cong \overline{Y'X}\)[/tex]
Congruency between [tex]\(\overline{YX}\)[/tex] and [tex]\(\overline{Y'X'}\)[/tex] would imply that the lengths of these segments are identical. However, due to the dilation by a factor of [tex]\(\frac{1}{2}\)[/tex], the corresponding segments in [tex]\(\triangle X'Y'Z'\)[/tex] are half the length of the original segments in [tex]\(\triangle XYZ\)[/tex]. Therefore, this statement is false.
#### Option 4: [tex]\(XZ = 2X'Z'\)[/tex]
Since the dilation reduces all side lengths by a factor of [tex]\(\frac{1}{2}\)[/tex], the new length [tex]\(X'Z'\)[/tex] is [tex]\(\frac{1}{2}\)[/tex] the length of [tex]\(XZ\)[/tex]. Hence,
[tex]\[ XZ = 2X'Z' \][/tex]
This statement is true.
#### Option 5: [tex]\(m \angle YXZ = 2 m \angle Y'X'Z'\)[/tex]
This statement suggests that the measure of angle [tex]\(YXZ\)[/tex] is twice the measure of angle [tex]\(Y'X'Z'\)[/tex]. Angles are preserved during reflections and dilations, which means that the measures of the corresponding angles remain the same. Therefore, this statement is false.
### Conclusion
Based on our analysis, the three statements that must be true are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(\angle XZY \cong \angle YZX\)[/tex]
3. [tex]\(XZ = 2X'Z'\)[/tex]
Thus, the correct choices are:
- [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
- [tex]\(\overline{XZ} = 2 \overline{X'Z'}\)[/tex]
- [tex]\(\angle XZY \cong \angle YZX\)[/tex]
Hence, the final answer is:
[tex]\[ (1, 2, 4) \][/tex]
