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Sagot :
Certainly! Let’s analyze which statements are true based on the transformations of [tex]\(\triangle XYZ\)[/tex] to [tex]\(\triangle X'Y'Z'\)[/tex].
The given transformations are:
1. Reflection over a vertical line.
2. Dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex].
Now, let's identify which properties are preserved by these transformations and apply them one by one.
Step-by-Step Analysis:
1. Similarity of Triangles:
- The dilation by a scale factor ([tex]\(\frac{1}{2}\)[/tex]) changes the size of the triangle but not its shape, ensuring that corresponding angles remain equal and sides are proportional.
- Thus, [tex]\(\triangle XYZ\)[/tex] is similar to [tex]\(\triangle X'Y'Z'\)[/tex] because dilation preserves the angles and the ratios of the corresponding side lengths.
- Conclusion: [tex]\(\triangle XYZ \sim \(\triangle X'Y'Z'\)[/tex]. This means that the statement [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex] is true.
2. Congruence of Angles:
- A reflection does not affect the measures of the angles.
- After dilation, the angles of the triangle remain unchanged.
- However, there is no guarantee that [tex]\(\angle XZ Y\)[/tex] is congruent to [tex]\(\angle Y'Z'X\)[/tex] based only on the given transformations because we do not know their positions relative to each other after reflection.
- Conclusion: [tex]\(\angle XZ Y \cong \angle Y'Z'X\)[/tex] is not necessarily true.
3. Congruence of Sides:
- The dilation with a scale factor of [tex]\(\frac{1}{2}\)[/tex] changes all the side lengths by the same proportion.
- This means each side length of the resulting [tex]\(\triangle X'Y'Z'\)[/tex] is half the corresponding side length of [tex]\(\triangle XYZ\)[/tex].
- Therefore, [tex]\(\overline{YX}\)[/tex] is not congruent to [tex]\(\overline{Y'X'}\)[/tex] because [tex]\(\overline{YX} = 2 \cdot \overline{Y'X'}\)[/tex].
- Conclusion: [tex]\(\overline{YX} \cong \overline{Y'X'}\)[/tex] is false.
4. Proportionality of Side Lengths:
- Given the scale factor of [tex]\(\frac{1}{2}\)[/tex], a side length in [tex]\(\triangle XYZ\)[/tex] will be twice the corresponding side length in the [tex]\(\triangle X'Y'Z'\)[/tex].
- Particularly, [tex]\(XZ = 2X'Z'\)[/tex] because the original side is twice the length of the corresponding side in the scaled triangle.
- Conclusion: [tex]\(XZ = 2X'Z'\)[/tex] is true.
5. Proportionality of Angles:
- Angles are not affected by dilation; they remain the same.
- Thus, [tex]\(\angle YXZ\)[/tex] will not be twice [tex]\(\angle Y'X'Z'\)[/tex] since dilation does not change the measures of the angles.
- Conclusion: [tex]\(m\angle YXZ \neq 2m\angle Y'X'Z'\)[/tex] is false.
True Statements:
Therefore, the statements that must be true are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(XZ = 2X'Z'\)[/tex]
These correspond to the answer (1, 4).
The given transformations are:
1. Reflection over a vertical line.
2. Dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex].
Now, let's identify which properties are preserved by these transformations and apply them one by one.
Step-by-Step Analysis:
1. Similarity of Triangles:
- The dilation by a scale factor ([tex]\(\frac{1}{2}\)[/tex]) changes the size of the triangle but not its shape, ensuring that corresponding angles remain equal and sides are proportional.
- Thus, [tex]\(\triangle XYZ\)[/tex] is similar to [tex]\(\triangle X'Y'Z'\)[/tex] because dilation preserves the angles and the ratios of the corresponding side lengths.
- Conclusion: [tex]\(\triangle XYZ \sim \(\triangle X'Y'Z'\)[/tex]. This means that the statement [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex] is true.
2. Congruence of Angles:
- A reflection does not affect the measures of the angles.
- After dilation, the angles of the triangle remain unchanged.
- However, there is no guarantee that [tex]\(\angle XZ Y\)[/tex] is congruent to [tex]\(\angle Y'Z'X\)[/tex] based only on the given transformations because we do not know their positions relative to each other after reflection.
- Conclusion: [tex]\(\angle XZ Y \cong \angle Y'Z'X\)[/tex] is not necessarily true.
3. Congruence of Sides:
- The dilation with a scale factor of [tex]\(\frac{1}{2}\)[/tex] changes all the side lengths by the same proportion.
- This means each side length of the resulting [tex]\(\triangle X'Y'Z'\)[/tex] is half the corresponding side length of [tex]\(\triangle XYZ\)[/tex].
- Therefore, [tex]\(\overline{YX}\)[/tex] is not congruent to [tex]\(\overline{Y'X'}\)[/tex] because [tex]\(\overline{YX} = 2 \cdot \overline{Y'X'}\)[/tex].
- Conclusion: [tex]\(\overline{YX} \cong \overline{Y'X'}\)[/tex] is false.
4. Proportionality of Side Lengths:
- Given the scale factor of [tex]\(\frac{1}{2}\)[/tex], a side length in [tex]\(\triangle XYZ\)[/tex] will be twice the corresponding side length in the [tex]\(\triangle X'Y'Z'\)[/tex].
- Particularly, [tex]\(XZ = 2X'Z'\)[/tex] because the original side is twice the length of the corresponding side in the scaled triangle.
- Conclusion: [tex]\(XZ = 2X'Z'\)[/tex] is true.
5. Proportionality of Angles:
- Angles are not affected by dilation; they remain the same.
- Thus, [tex]\(\angle YXZ\)[/tex] will not be twice [tex]\(\angle Y'X'Z'\)[/tex] since dilation does not change the measures of the angles.
- Conclusion: [tex]\(m\angle YXZ \neq 2m\angle Y'X'Z'\)[/tex] is false.
True Statements:
Therefore, the statements that must be true are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(XZ = 2X'Z'\)[/tex]
These correspond to the answer (1, 4).
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