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Sagot :
To determine which statements must be true for the given transformations of [tex]$\triangle XYZ$[/tex] resulting in [tex]$\triangle X'Y'Z'$[/tex], let’s carefully analyze each option:
1. Triangles with a dilation transformation are always similar:
When a triangle is dilated, its shape remains the same, but its size changes proportionally. This means that corresponding angles of the original triangle and the dilated triangle are congruent, and the lengths of the sides are proportional. Therefore, the triangles are similar.
[tex]\[ \triangle XYZ \sim \triangle X'Y'Z' \][/tex]
This statement is true.
2. Reflection preserves angle measures, so corresponding angles remain congruent:
When a triangle is reflected over a line, the angles of the triangle remain the same. Thus, the measures of the corresponding angles in [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X'Y'Z'$[/tex] after reflection and dilation are congruent.
[tex]\[ \angle XZY \cong \angle Y'Z'X' \][/tex]
This statement is true.
3. Reflection does not necessarily preserve segment lengths, and dilation scales segments by the scale factor:
A reflection will preserve lengths, but subsequent dilation changes all side lengths by the scale factor. Given that the dilation factor here is [tex]$\frac{1}{2}$[/tex], the sides of [tex]$\triangle X'Y'Z'$[/tex] will be half the length of the corresponding sides in [tex]$\triangle XYZ$[/tex]. Thus, [tex]\( \overline{YX} \)[/tex] does not equal [tex]\( \overline{Y'X'} \)[/tex] unless the lengths are zero, which they are not.
This statement is false.
4. Given dilation scale factor is [tex]$\frac{1}{2}$[/tex], so the original segment length is 2 times the new one:
Since the dilation scale factor is [tex]$\frac{1}{2}$[/tex], the length of any segment in [tex]$\triangle XYZ$[/tex] will be twice the length of the corresponding segment in [tex]$\triangle X'Y'Z'$[/tex]. Hence, if we take [tex]$XZ$[/tex] as the segment, we get:
[tex]\[ XZ = 2 \times X'Z' \][/tex]
This statement is true.
5. Angle measures remain unchanged in reflection and dilation:
Both reflection and dilation do not alter the angle measures. Therefore, the measure of angle [tex]\( \angle YXZ \)[/tex] in the original triangle will be the same in the dilated triangle [tex]\( \angle Y'X'Z' \)[/tex]. The statement saying, [tex]\( m\angle YXZ = 2 \times m \angle Y'X'Z' \)[/tex], indicates that the angle in the original triangle would be twice the angle in the dilated triangle, which is incorrect because angles are preserved.
This statement is false.
Based on this analysis, the true statements 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 the first, second, and fourth statements:
[tex]\[ (1, 1, 1) \][/tex]
1. Triangles with a dilation transformation are always similar:
When a triangle is dilated, its shape remains the same, but its size changes proportionally. This means that corresponding angles of the original triangle and the dilated triangle are congruent, and the lengths of the sides are proportional. Therefore, the triangles are similar.
[tex]\[ \triangle XYZ \sim \triangle X'Y'Z' \][/tex]
This statement is true.
2. Reflection preserves angle measures, so corresponding angles remain congruent:
When a triangle is reflected over a line, the angles of the triangle remain the same. Thus, the measures of the corresponding angles in [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X'Y'Z'$[/tex] after reflection and dilation are congruent.
[tex]\[ \angle XZY \cong \angle Y'Z'X' \][/tex]
This statement is true.
3. Reflection does not necessarily preserve segment lengths, and dilation scales segments by the scale factor:
A reflection will preserve lengths, but subsequent dilation changes all side lengths by the scale factor. Given that the dilation factor here is [tex]$\frac{1}{2}$[/tex], the sides of [tex]$\triangle X'Y'Z'$[/tex] will be half the length of the corresponding sides in [tex]$\triangle XYZ$[/tex]. Thus, [tex]\( \overline{YX} \)[/tex] does not equal [tex]\( \overline{Y'X'} \)[/tex] unless the lengths are zero, which they are not.
This statement is false.
4. Given dilation scale factor is [tex]$\frac{1}{2}$[/tex], so the original segment length is 2 times the new one:
Since the dilation scale factor is [tex]$\frac{1}{2}$[/tex], the length of any segment in [tex]$\triangle XYZ$[/tex] will be twice the length of the corresponding segment in [tex]$\triangle X'Y'Z'$[/tex]. Hence, if we take [tex]$XZ$[/tex] as the segment, we get:
[tex]\[ XZ = 2 \times X'Z' \][/tex]
This statement is true.
5. Angle measures remain unchanged in reflection and dilation:
Both reflection and dilation do not alter the angle measures. Therefore, the measure of angle [tex]\( \angle YXZ \)[/tex] in the original triangle will be the same in the dilated triangle [tex]\( \angle Y'X'Z' \)[/tex]. The statement saying, [tex]\( m\angle YXZ = 2 \times m \angle Y'X'Z' \)[/tex], indicates that the angle in the original triangle would be twice the angle in the dilated triangle, which is incorrect because angles are preserved.
This statement is false.
Based on this analysis, the true statements 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 the first, second, and fourth statements:
[tex]\[ (1, 1, 1) \][/tex]
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