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Sagot :
To analyze the given transformations and determine which statements must be true, let's go through each statement step-by-step:
1. Statement 1: [tex]$\triangle XYZ \sim \triangle X'Y'Z$[/tex]
- Similarity of Triangles: When a triangle undergoes a reflection, it remains similar to the original triangle because their angles are congruent and the shapes are the same. Dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] will also result in similar triangles because it preserves the shape and angles, even though the side lengths are halved.
- Conclusion: This statement is true.
2. Statement 2: [tex]$\angle XZY \cong \angle YZX$[/tex]
- Congruency of Angles: Reflection does not change the measures of angles within the triangle. Dilation preserves angles as well.
- Conclusion: This statement is true.
3. Statement 3: [tex]$\overline{YX} \approx \overline{Y'X'}$[/tex]
- Segment Lengths: Dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] changes the lengths of the sides. Specifically, the sides of [tex]$\triangle X'Y'Z'$[/tex] are half the lengths of the corresponding sides in [tex]$\triangle XYZ$[/tex]. Therefore, [tex]$\overline{YX}$[/tex] is not approximately equal to [tex]$\overline{Y'X'}$[/tex] because [tex]$\overline{YX'} = \frac{1}{2}\overline{YX}$[/tex].
- Conclusion: This statement is false.
4. Statement 4: [tex]$XZ = 2 X'Z'$[/tex]
- Relation of Side Lengths: Given the scale factor of [tex]$\frac{1}{2}$[/tex] for dilation, sides of [tex]$\triangle XYZ$[/tex] will be twice as long as the corresponding sides in [tex]$\triangle X'Y'Z'$[/tex]. Therefore, [tex]$XZ = 2 X'Z'$[/tex] holds true.
- Conclusion: This statement is true.
5. Statement 5: [tex]$m\angle YXZ = 2m\angle Y'X'Z'$[/tex]
- Scalability of Angles: Reflections and dilations preserve the measure of angles; they do not double or halve the angles. Hence, [tex]$m\angle YXZ$[/tex] should be equal to [tex]$m\angle Y'X'Z'$[/tex].
- Conclusion: This statement is false.
Final Conclusion: The true statements are:
- [tex]$\triangle XYZ \sim \triangle X'Y'Z$[/tex]
- [tex]$\angle XZY \cong \angle YZX$[/tex]
- [tex]$XZ = 2X'Z'$[/tex]
Therefore, the correct options to select are:
1, 2, and 4.
1. Statement 1: [tex]$\triangle XYZ \sim \triangle X'Y'Z$[/tex]
- Similarity of Triangles: When a triangle undergoes a reflection, it remains similar to the original triangle because their angles are congruent and the shapes are the same. Dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] will also result in similar triangles because it preserves the shape and angles, even though the side lengths are halved.
- Conclusion: This statement is true.
2. Statement 2: [tex]$\angle XZY \cong \angle YZX$[/tex]
- Congruency of Angles: Reflection does not change the measures of angles within the triangle. Dilation preserves angles as well.
- Conclusion: This statement is true.
3. Statement 3: [tex]$\overline{YX} \approx \overline{Y'X'}$[/tex]
- Segment Lengths: Dilation by a scale factor of [tex]$\frac{1}{2}$[/tex] changes the lengths of the sides. Specifically, the sides of [tex]$\triangle X'Y'Z'$[/tex] are half the lengths of the corresponding sides in [tex]$\triangle XYZ$[/tex]. Therefore, [tex]$\overline{YX}$[/tex] is not approximately equal to [tex]$\overline{Y'X'}$[/tex] because [tex]$\overline{YX'} = \frac{1}{2}\overline{YX}$[/tex].
- Conclusion: This statement is false.
4. Statement 4: [tex]$XZ = 2 X'Z'$[/tex]
- Relation of Side Lengths: Given the scale factor of [tex]$\frac{1}{2}$[/tex] for dilation, sides of [tex]$\triangle XYZ$[/tex] will be twice as long as the corresponding sides in [tex]$\triangle X'Y'Z'$[/tex]. Therefore, [tex]$XZ = 2 X'Z'$[/tex] holds true.
- Conclusion: This statement is true.
5. Statement 5: [tex]$m\angle YXZ = 2m\angle Y'X'Z'$[/tex]
- Scalability of Angles: Reflections and dilations preserve the measure of angles; they do not double or halve the angles. Hence, [tex]$m\angle YXZ$[/tex] should be equal to [tex]$m\angle Y'X'Z'$[/tex].
- Conclusion: This statement is false.
Final Conclusion: The true statements are:
- [tex]$\triangle XYZ \sim \triangle X'Y'Z$[/tex]
- [tex]$\angle XZY \cong \angle YZX$[/tex]
- [tex]$XZ = 2X'Z'$[/tex]
Therefore, the correct options to select are:
1, 2, and 4.
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