Answer :
Let's carefully analyze the given transformations and their effects on the properties of [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex].
1. Reflection Over a Vertical Line:
- This transformation preserves the shape and size of the triangle but reverses its orientation. Angles stay the same, and lengths are unchanged.
2. Dilation by a Scale Factor of [tex]$\frac{1}{2}$[/tex]:
- This transformation reduces all distances by a factor of [tex]$\frac{1}{2}$[/tex]. The angles remain the same, but the side lengths are halved.
Given these details, let's evaluate the options:
1. [tex]$\triangle XYZ \sim \triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex]:
- Similarity in geometry means that the triangles have the same shape, although not necessarily the same size. Angles are preserved, and side lengths are proportional. After the reflection and dilation, the shape is preserved, but the size is scaled down by [tex]$\frac{1}{2}$[/tex]. Hence, [tex]$\triangle XYZ$[/tex] is similar to [tex]$\triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex]. This statement is true.
2. [tex]$\angle XZY \simeq \angle Y^{\prime} Z^{\prime} X^{\prime}$[/tex]:
- Angles in a triangle remain unchanged during both reflection and dilation. Therefore, each corresponding angle in [tex]$\triangle XYZ$[/tex] remains equal to the corresponding angle in [tex]$\triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex]. This statement is true.
3. [tex]$\overline{YX} \approx \overline{Y^{\prime} X^{\prime}}$[/tex]:
- This notation typically indicates that the line segments are congruent. However, due to the dilation with a scale factor of [tex]$\frac{1}{2}$[/tex], the lengths of corresponding sides in [tex]$\triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex] will be half of those in [tex]$\triangle XYZ$[/tex]. Thus, [tex]$\overline{YX} \not\approx \overline{Y^{\prime} X^{\prime}}$[/tex]. This statement is false.
4. [tex]$XZ=2X^{\prime} Z^{\prime}$[/tex]:
- Dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] means that each side length in [tex]$\triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex] is half of the corresponding side length in [tex]$\triangle XYZ$[/tex]. Hence, [tex]$XZ$[/tex] in the original triangle is twice the length of [tex]$X^{\prime} Z^{\prime}$[/tex] in the dilated triangle. This statement is true.
5. [tex]$m \angle YXZ=2m \angle Y^{\prime} X^{\prime} Z^{\prime}$[/tex]:
- The measure of the angles does not change under dilation or reflection. Thus, [tex]$m \angle YXZ$[/tex] should be equal to [tex]$m \angle Y^{\prime} X^{\prime} Z^{\prime}$[/tex], not twice it. This statement is false.
Therefore, the three true statements are:
- [tex]$\triangle XYZ \sim \triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex]
- [tex]$\angle XZY \simeq \angle Y^{\prime} Z^{\prime} X^{\prime}$[/tex]
- [tex]$XZ=2X^{\prime} Z^{\prime}$[/tex]
The correct options are [1, 2, 4].
1. Reflection Over a Vertical Line:
- This transformation preserves the shape and size of the triangle but reverses its orientation. Angles stay the same, and lengths are unchanged.
2. Dilation by a Scale Factor of [tex]$\frac{1}{2}$[/tex]:
- This transformation reduces all distances by a factor of [tex]$\frac{1}{2}$[/tex]. The angles remain the same, but the side lengths are halved.
Given these details, let's evaluate the options:
1. [tex]$\triangle XYZ \sim \triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex]:
- Similarity in geometry means that the triangles have the same shape, although not necessarily the same size. Angles are preserved, and side lengths are proportional. After the reflection and dilation, the shape is preserved, but the size is scaled down by [tex]$\frac{1}{2}$[/tex]. Hence, [tex]$\triangle XYZ$[/tex] is similar to [tex]$\triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex]. This statement is true.
2. [tex]$\angle XZY \simeq \angle Y^{\prime} Z^{\prime} X^{\prime}$[/tex]:
- Angles in a triangle remain unchanged during both reflection and dilation. Therefore, each corresponding angle in [tex]$\triangle XYZ$[/tex] remains equal to the corresponding angle in [tex]$\triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex]. This statement is true.
3. [tex]$\overline{YX} \approx \overline{Y^{\prime} X^{\prime}}$[/tex]:
- This notation typically indicates that the line segments are congruent. However, due to the dilation with a scale factor of [tex]$\frac{1}{2}$[/tex], the lengths of corresponding sides in [tex]$\triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex] will be half of those in [tex]$\triangle XYZ$[/tex]. Thus, [tex]$\overline{YX} \not\approx \overline{Y^{\prime} X^{\prime}}$[/tex]. This statement is false.
4. [tex]$XZ=2X^{\prime} Z^{\prime}$[/tex]:
- Dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] means that each side length in [tex]$\triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex] is half of the corresponding side length in [tex]$\triangle XYZ$[/tex]. Hence, [tex]$XZ$[/tex] in the original triangle is twice the length of [tex]$X^{\prime} Z^{\prime}$[/tex] in the dilated triangle. This statement is true.
5. [tex]$m \angle YXZ=2m \angle Y^{\prime} X^{\prime} Z^{\prime}$[/tex]:
- The measure of the angles does not change under dilation or reflection. Thus, [tex]$m \angle YXZ$[/tex] should be equal to [tex]$m \angle Y^{\prime} X^{\prime} Z^{\prime}$[/tex], not twice it. This statement is false.
Therefore, the three true statements are:
- [tex]$\triangle XYZ \sim \triangle X^{\prime} Y^{\prime} Z^{\prime}$[/tex]
- [tex]$\angle XZY \simeq \angle Y^{\prime} Z^{\prime} X^{\prime}$[/tex]
- [tex]$XZ=2X^{\prime} Z^{\prime}$[/tex]
The correct options are [1, 2, 4].
