Answer :
Let's analyze the transformations applied to [tex]$\triangle XYZ$[/tex] to form [tex]$\triangle X'Y'Z'$[/tex] and determine what must be true about the resulting triangles. The two transformations are:
1. Reflection over a vertical line.
2. Dilation by a scale factor of [tex]$\frac{1}{2}$[/tex].
Given these transformations, let's discuss each of the provided options:
1. Similarity: [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]
- A reflection is an isometry, which means it preserves lengths and angles, making the two triangles congruent in terms of shape and size.
- A dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] scales all lengths by [tex]$\frac{1}{2}$[/tex] while preserving the angles.
- Since both transformations preserve angles and the dilation affects all side lengths proportionally, [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X'Y'Z'$[/tex] are similar. Thus, this statement is true.
2. Angle Equality: [tex]$\angle XZY = \angle Y'Z'X'$[/tex]
- Reflections preserve angles because they do not alter the amount of rotation between lines intersecting at a point.
- Additionally, dilation also preserves the measure of the angles.
- Therefore, the angles in [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X'Y'Z'$[/tex] remain equal. This statement is true.
3. Equality of Sides: [tex]$\overline{YX} = \overline{Y'X'}$[/tex]
- A reflection by itself would preserve the lengths of sides, making [tex]$\overline{YX} = \overline{Y'X'}$[/tex] true for that transformation.
- However, the dilation by [tex]$\frac{1}{2}$[/tex] scales every length in [tex]$\triangle XYZ$[/tex] by [tex]$\frac{1}{2}$[/tex].
- Hence, [tex]$\overline{YX}$[/tex] cannot be equal to [tex]$\overline{Y'X'}$[/tex] after scaling. This statement is false.
4. Length Relation: [tex]$XZ = 2 X'Z'$[/tex]
- After reflecting, the lengths in [tex]$\triangle XYZ$[/tex] are not altered.
- With the dilation by [tex]$\frac{1}{2}$[/tex], each length, including [tex]$\overline{XZ}$[/tex], is scaled by [tex]$\frac{1}{2}$[/tex].
- Therefore, the original length [tex]$XZ$[/tex] will be twice the length of [tex]$X'Z'$[/tex] formed after dilation. This statement is true.
5. Angle Magnitude: [tex]$m\angle YXZ = 2 m\angle Y'X'Z'$[/tex]
- Reflections and dilations both preserve the measures of angles.
- Consequently, the original and the final angles must be equal, not double. This statement is false.
Based on these analyses, the three correct options are:
- [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]
- [tex]$\angle XZY = \angle Y'Z'X'$[/tex]
- [tex]$XZ = 2 X'Z'$[/tex]
1. Reflection over a vertical line.
2. Dilation by a scale factor of [tex]$\frac{1}{2}$[/tex].
Given these transformations, let's discuss each of the provided options:
1. Similarity: [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]
- A reflection is an isometry, which means it preserves lengths and angles, making the two triangles congruent in terms of shape and size.
- A dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] scales all lengths by [tex]$\frac{1}{2}$[/tex] while preserving the angles.
- Since both transformations preserve angles and the dilation affects all side lengths proportionally, [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X'Y'Z'$[/tex] are similar. Thus, this statement is true.
2. Angle Equality: [tex]$\angle XZY = \angle Y'Z'X'$[/tex]
- Reflections preserve angles because they do not alter the amount of rotation between lines intersecting at a point.
- Additionally, dilation also preserves the measure of the angles.
- Therefore, the angles in [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X'Y'Z'$[/tex] remain equal. This statement is true.
3. Equality of Sides: [tex]$\overline{YX} = \overline{Y'X'}$[/tex]
- A reflection by itself would preserve the lengths of sides, making [tex]$\overline{YX} = \overline{Y'X'}$[/tex] true for that transformation.
- However, the dilation by [tex]$\frac{1}{2}$[/tex] scales every length in [tex]$\triangle XYZ$[/tex] by [tex]$\frac{1}{2}$[/tex].
- Hence, [tex]$\overline{YX}$[/tex] cannot be equal to [tex]$\overline{Y'X'}$[/tex] after scaling. This statement is false.
4. Length Relation: [tex]$XZ = 2 X'Z'$[/tex]
- After reflecting, the lengths in [tex]$\triangle XYZ$[/tex] are not altered.
- With the dilation by [tex]$\frac{1}{2}$[/tex], each length, including [tex]$\overline{XZ}$[/tex], is scaled by [tex]$\frac{1}{2}$[/tex].
- Therefore, the original length [tex]$XZ$[/tex] will be twice the length of [tex]$X'Z'$[/tex] formed after dilation. This statement is true.
5. Angle Magnitude: [tex]$m\angle YXZ = 2 m\angle Y'X'Z'$[/tex]
- Reflections and dilations both preserve the measures of angles.
- Consequently, the original and the final angles must be equal, not double. This statement is false.
Based on these analyses, the three correct options are:
- [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex]
- [tex]$\angle XZY = \angle Y'Z'X'$[/tex]
- [tex]$XZ = 2 X'Z'$[/tex]
