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Sagot :
Let's analyze each statement about the triangles [tex]\( \triangle XYZ \)[/tex] and [tex]\( \triangle X'Y'Z' \)[/tex] after reflection over a vertical line and dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex]:
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]:
Triangles are similar if they have the same shape but not necessarily the same size. Similarity is preserved under reflections and dilations because these transformations do not alter the relative angles between corresponding sides. Hence, [tex]\( \triangle XYZ \)[/tex] is similar to [tex]\( \triangle X'Y'Z' \)[/tex].
2. [tex]\( \angle XZY \cong \angle Y'Z'X \)[/tex]:
Angle congruence is preserved under both reflection and dilation. This means that each angle in [tex]\( \triangle XYZ \)[/tex] corresponds to an equal angle in [tex]\( \triangle X'Y'Z' \)[/tex]. Thus, [tex]\( \angle XZY \cong \angle Y'Z'X \)[/tex].
3. [tex]\( \overline{YX} \approx \overline{Y'X'} \)[/tex] (YX is congruent to Y'X'):
Congruence of segments implies that they have the same length. However, dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] changes the lengths of all sides of [tex]\( \triangle XYZ \)[/tex] by the same factor. This means that [tex]\( \overline{YX} \)[/tex] will not be congruent to [tex]\( \overline{Y'X'} \)[/tex], as [tex]\( \overline{Y'X'} \)[/tex] will be half the length of [tex]\( \overline{YX} \)[/tex].
4. [tex]\( XZ = 2X'Z' \)[/tex]:
Given the dilation factor of [tex]\(\frac{1}{2}\)[/tex], the sides of [tex]\( \triangle X'Y'Z' \)[/tex] are half the lengths of the corresponding sides in [tex]\( \triangle XYZ \)[/tex]. Therefore, [tex]\( XZ \)[/tex] is indeed twice the length of [tex]\( X'Z' \)[/tex].
5. [tex]\( m \angle YXZ = 2m \angle Y'X'Z' \)[/tex]:
Since the dilation does not affect the angles, the measures of corresponding angles in [tex]\( \triangle XYZ \)[/tex] and [tex]\( \triangle X'Y'Z' \)[/tex] remain equal. Hence, the measure of [tex]\( \angle YXZ \)[/tex] is not twice but equal to the measure of [tex]\( \angle Y'X'Z' \)[/tex].
The true statements from the above analysis are:
- [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]
- [tex]\( \angle XZY \cong \angle Y'Z'X \)[/tex]
- [tex]\( XZ = 2X'Z' \)[/tex]
Thus, the correct options are:
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]
2. [tex]\( \angle XZY \cong \angle Y'Z'X \)[/tex]
3. [tex]\( XZ = 2X'Z' \)[/tex]
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]:
Triangles are similar if they have the same shape but not necessarily the same size. Similarity is preserved under reflections and dilations because these transformations do not alter the relative angles between corresponding sides. Hence, [tex]\( \triangle XYZ \)[/tex] is similar to [tex]\( \triangle X'Y'Z' \)[/tex].
2. [tex]\( \angle XZY \cong \angle Y'Z'X \)[/tex]:
Angle congruence is preserved under both reflection and dilation. This means that each angle in [tex]\( \triangle XYZ \)[/tex] corresponds to an equal angle in [tex]\( \triangle X'Y'Z' \)[/tex]. Thus, [tex]\( \angle XZY \cong \angle Y'Z'X \)[/tex].
3. [tex]\( \overline{YX} \approx \overline{Y'X'} \)[/tex] (YX is congruent to Y'X'):
Congruence of segments implies that they have the same length. However, dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] changes the lengths of all sides of [tex]\( \triangle XYZ \)[/tex] by the same factor. This means that [tex]\( \overline{YX} \)[/tex] will not be congruent to [tex]\( \overline{Y'X'} \)[/tex], as [tex]\( \overline{Y'X'} \)[/tex] will be half the length of [tex]\( \overline{YX} \)[/tex].
4. [tex]\( XZ = 2X'Z' \)[/tex]:
Given the dilation factor of [tex]\(\frac{1}{2}\)[/tex], the sides of [tex]\( \triangle X'Y'Z' \)[/tex] are half the lengths of the corresponding sides in [tex]\( \triangle XYZ \)[/tex]. Therefore, [tex]\( XZ \)[/tex] is indeed twice the length of [tex]\( X'Z' \)[/tex].
5. [tex]\( m \angle YXZ = 2m \angle Y'X'Z' \)[/tex]:
Since the dilation does not affect the angles, the measures of corresponding angles in [tex]\( \triangle XYZ \)[/tex] and [tex]\( \triangle X'Y'Z' \)[/tex] remain equal. Hence, the measure of [tex]\( \angle YXZ \)[/tex] is not twice but equal to the measure of [tex]\( \angle Y'X'Z' \)[/tex].
The true statements from the above analysis are:
- [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]
- [tex]\( \angle XZY \cong \angle Y'Z'X \)[/tex]
- [tex]\( XZ = 2X'Z' \)[/tex]
Thus, the correct options are:
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]
2. [tex]\( \angle XZY \cong \angle Y'Z'X \)[/tex]
3. [tex]\( XZ = 2X'Z' \)[/tex]
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