Answer :
Let's analyze the transformations applied to [tex]\(\triangle X Y Z\)[/tex]:
1. Reflection over a vertical line:
- This transformation flips the triangle across a vertical axis. While this changes the orientation of the triangle, it does not change the size or shape of the triangle.
2. Dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex]:
- Dilation reduces the size of the triangle by a factor of [tex]\(\frac{1}{2}\)[/tex]. The shape and proportion of the triangle remain unchanged, but every side length is half of the corresponding side length in the original triangle.
Given these transformations, we can determine the following properties of [tex]\(\triangle X Y Z\)[/tex] and [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex]:
1. Similarity of triangles:
- Since dilation does not alter angles, the resulting triangle [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] is similar to the original triangle [tex]\(\triangle X Y Z\)[/tex]. The similarity ratio is [tex]\(\frac{1}{2}\)[/tex].
- [tex]\(\triangle X Y Z \sim \triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] is true.
2. Equality of angles:
- Reflection and dilation do not affect the angles of the triangle. Thus, the angles of [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] remain equal to the corresponding angles of [tex]\(\triangle X Y Z\)[/tex].
- Therefore, [tex]\(\angle X Z Y = \angle Y^{\prime} Z^{\prime} X^{\prime}\)[/tex] is true.
3. Equality of corresponding side lengths:
- Since dilation with a scale factor of [tex]\(\frac{1}{2}\)[/tex] reduces all the lengths by half, the corresponding side lengths of [tex]\(\triangle X Y Z\)[/tex] and [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] are related by the same factor.
- [tex]\(\overline{Y X}=\overline{Y^{\prime} X^{\prime}}\)[/tex] is false because [tex]\(\overline{Y' X'} = \frac{1}{2} \overline{Y X}\)[/tex].
4. Relationship between the original and dilated side lengths:
- If a side in the original triangle is [tex]\(X Z\)[/tex], its corresponding side in [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] is [tex]\(\frac{1}{2} X Z\)[/tex].
- Hence, [tex]\(X Z = 2 X^{\prime} Z^{\prime}\)[/tex] is true.
5. Comparison of angles' measures with a factor:
- Since dilation and reflection do not affect angles, the measure of the angles remains unchanged.
- Therefore, [tex]\(m \angle Y X Z = 2 m \angle Y^{\prime} X^{\prime} Z^{\prime}\)[/tex] is false because the angles are not doubled but remain the same.
To summarize, the three true statements 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]
These properties result from the reflection and dilation transformations applied to [tex]\(\triangle X Y Z\)[/tex].
1. Reflection over a vertical line:
- This transformation flips the triangle across a vertical axis. While this changes the orientation of the triangle, it does not change the size or shape of the triangle.
2. Dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex]:
- Dilation reduces the size of the triangle by a factor of [tex]\(\frac{1}{2}\)[/tex]. The shape and proportion of the triangle remain unchanged, but every side length is half of the corresponding side length in the original triangle.
Given these transformations, we can determine the following properties of [tex]\(\triangle X Y Z\)[/tex] and [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex]:
1. Similarity of triangles:
- Since dilation does not alter angles, the resulting triangle [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] is similar to the original triangle [tex]\(\triangle X Y Z\)[/tex]. The similarity ratio is [tex]\(\frac{1}{2}\)[/tex].
- [tex]\(\triangle X Y Z \sim \triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] is true.
2. Equality of angles:
- Reflection and dilation do not affect the angles of the triangle. Thus, the angles of [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] remain equal to the corresponding angles of [tex]\(\triangle X Y Z\)[/tex].
- Therefore, [tex]\(\angle X Z Y = \angle Y^{\prime} Z^{\prime} X^{\prime}\)[/tex] is true.
3. Equality of corresponding side lengths:
- Since dilation with a scale factor of [tex]\(\frac{1}{2}\)[/tex] reduces all the lengths by half, the corresponding side lengths of [tex]\(\triangle X Y Z\)[/tex] and [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] are related by the same factor.
- [tex]\(\overline{Y X}=\overline{Y^{\prime} X^{\prime}}\)[/tex] is false because [tex]\(\overline{Y' X'} = \frac{1}{2} \overline{Y X}\)[/tex].
4. Relationship between the original and dilated side lengths:
- If a side in the original triangle is [tex]\(X Z\)[/tex], its corresponding side in [tex]\(\triangle X^{\prime} Y^{\prime} Z^{\prime}\)[/tex] is [tex]\(\frac{1}{2} X Z\)[/tex].
- Hence, [tex]\(X Z = 2 X^{\prime} Z^{\prime}\)[/tex] is true.
5. Comparison of angles' measures with a factor:
- Since dilation and reflection do not affect angles, the measure of the angles remains unchanged.
- Therefore, [tex]\(m \angle Y X Z = 2 m \angle Y^{\prime} X^{\prime} Z^{\prime}\)[/tex] is false because the angles are not doubled but remain the same.
To summarize, the three true statements 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]
These properties result from the reflection and dilation transformations applied to [tex]\(\triangle X Y Z\)[/tex].
