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Sagot :
Certainly! Let's break down the problem and determine which of the given statements are true when [tex]$\triangle XYZ$[/tex] is reflected over a vertical line and then dilated by a scale factor of [tex]\(\frac{1}{2}\)[/tex], resulting in [tex]\(\triangle X'Y'Z'\)[/tex].
1. Similarity of Triangles:
[tex]\[ \triangle XYZ \sim \triangle X'Y'Z' \][/tex]
When a triangle is reflected and then scaled by any multiplier, the shape of the triangle does not change; only its size does. Reflection and dilation preserve the angles and the ratios of the sides. Therefore, [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle X'Y'Z'\)[/tex] are similar triangles.
[tex]\[ \text{True} \][/tex]
2. Angle Congruence:
[tex]\[ \angle XZY \cong \angle Y'Z'X' \][/tex]
Reflection over a line and dilation by a scale factor do not affect the measures of the angles of the triangle. Therefore, corresponding angles remain congruent.
[tex]\[ \text{True} \][/tex]
3. Approximation of segments:
[tex]\[ \overline{YX} \approx \overline{Y'X} \][/tex]
Since the transformation involves a dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex] and not an exact congruence transformation (where lengths would remain the same), this statement about the lengths being approximately equal is incorrect.
[tex]\[ \text{False} \][/tex]
4. Scale of Side Lengths:
[tex]\[ XZ = 2X'Z' \][/tex]
Given the scale factor of [tex]\(\frac{1}{2}\)[/tex], the sides of [tex]\(\triangle X'Y'Z'\)[/tex] are half the length of the corresponding sides in [tex]\(\triangle XYZ\)[/tex]. Hence, [tex]\(XZ\)[/tex] would be twice the length of [tex]\(X'Z'\)[/tex].
[tex]\[ \text{True} \][/tex]
5. Scaling Angle Measures:
[tex]\[ m \angle YXZ=2 m \angle YY'X'Z' \][/tex]
Angle measures are invariant under reflection and dilation. This means that if the measure of an angle in [tex]\(\triangle XYZ\)[/tex] is [tex]\(m\angle YXZ\)[/tex], it is the same as the corresponding angle in [tex]\(\triangle X'Y'Z'\)[/tex]. They do not double or halve.
[tex]\[ \text{False} \][/tex]
So, the three true statements are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(\angle XZY \cong \angle Y'Z'X'\)[/tex]
4. [tex]\(XZ = 2 X'Z'\)[/tex]
Hence, the results are:
[tex]\[ \boxed{(\text{True}, \text{True}, \text{False}, \text{True}, \text{False})} \][/tex]
1. Similarity of Triangles:
[tex]\[ \triangle XYZ \sim \triangle X'Y'Z' \][/tex]
When a triangle is reflected and then scaled by any multiplier, the shape of the triangle does not change; only its size does. Reflection and dilation preserve the angles and the ratios of the sides. Therefore, [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle X'Y'Z'\)[/tex] are similar triangles.
[tex]\[ \text{True} \][/tex]
2. Angle Congruence:
[tex]\[ \angle XZY \cong \angle Y'Z'X' \][/tex]
Reflection over a line and dilation by a scale factor do not affect the measures of the angles of the triangle. Therefore, corresponding angles remain congruent.
[tex]\[ \text{True} \][/tex]
3. Approximation of segments:
[tex]\[ \overline{YX} \approx \overline{Y'X} \][/tex]
Since the transformation involves a dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex] and not an exact congruence transformation (where lengths would remain the same), this statement about the lengths being approximately equal is incorrect.
[tex]\[ \text{False} \][/tex]
4. Scale of Side Lengths:
[tex]\[ XZ = 2X'Z' \][/tex]
Given the scale factor of [tex]\(\frac{1}{2}\)[/tex], the sides of [tex]\(\triangle X'Y'Z'\)[/tex] are half the length of the corresponding sides in [tex]\(\triangle XYZ\)[/tex]. Hence, [tex]\(XZ\)[/tex] would be twice the length of [tex]\(X'Z'\)[/tex].
[tex]\[ \text{True} \][/tex]
5. Scaling Angle Measures:
[tex]\[ m \angle YXZ=2 m \angle YY'X'Z' \][/tex]
Angle measures are invariant under reflection and dilation. This means that if the measure of an angle in [tex]\(\triangle XYZ\)[/tex] is [tex]\(m\angle YXZ\)[/tex], it is the same as the corresponding angle in [tex]\(\triangle X'Y'Z'\)[/tex]. They do not double or halve.
[tex]\[ \text{False} \][/tex]
So, the three true statements are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(\angle XZY \cong \angle Y'Z'X'\)[/tex]
4. [tex]\(XZ = 2 X'Z'\)[/tex]
Hence, the results are:
[tex]\[ \boxed{(\text{True}, \text{True}, \text{False}, \text{True}, \text{False})} \][/tex]
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