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To determine which statements must be true about the triangles [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle X'Y'Z'\)[/tex] after the transformations (reflection and dilation), we should analyze the geometric properties preserved under these transformations.
1. Reflection:
- A reflection over a vertical line flips the figure horizontally but preserves the shape and size. Therefore, the reflected triangle [tex]\(\triangle XYZ\)[/tex] will still be congruent and similar to [tex]\(\triangle XYZ\)[/tex]. Angles remain the same.
2. Dilation by a Scale Factor of [tex]\(\frac{1}{2}\)[/tex]:
- Dilation changes the size of the figure by a specific factor while maintaining the shape. With a scale factor of [tex]\(\frac{1}{2}\)[/tex], each side of the triangle is halved, but the angles remain unchanged because dilation is a similarity transformation.
Let's evaluate each option in light of the transformations:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]:
- Both reflection and dilation are similarity transformations. Since both transformations (reflection and dilation) occur, the resulting triangles [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle X'Y'Z'\)[/tex] are similar. Therefore, this statement is true.
2. [tex]\(\angle XZY \cong \angle Y'Z'X'\)[/tex]:
- Angles remain unchanged in both reflection and dilation. Hence, the corresponding angles [tex]\(\angle XZY\)[/tex] and [tex]\(\angle Y'Z'X'\)[/tex] must be congruent. Therefore, this statement is true.
3. [tex]\(\overline{YX} \cong \overline{Y'X'}\)[/tex]:
- Under dilation with a scale factor of [tex]\(\frac{1}{2}\)[/tex], the corresponding sides are halved. Therefore, [tex]\(\overline{YX}\)[/tex] is not congruent to [tex]\(\overline{Y'X'}\)[/tex]. Hence, this statement is false.
4. [tex]\(XZ = 2X'Z'\)[/tex]:
- After dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex], the new side lengths are half of the original side lengths. Therefore, the original side length [tex]\(XZ\)[/tex] is twice the length of the new side [tex]\(X'Z'\)[/tex]. Hence, this statement is true.
5. [tex]\(m\angle YXZ = 2 m\angle Y'X'Z'\)[/tex]:
- The measure of angles does not change under dilation. Therefore, [tex]\(m\angle YXZ\)[/tex] is equal to [tex]\(m\angle Y'X'Z'\)[/tex], not twice. Hence, this statement is false.
To summarize, three statements that must be true are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(\angle XZY \cong \angle Y'Z'X'\)[/tex]
4. [tex]\(XZ = 2X'Z'\)[/tex]
These reflect the relationships and properties preserved during the reflection and dilation transformations.
1. Reflection:
- A reflection over a vertical line flips the figure horizontally but preserves the shape and size. Therefore, the reflected triangle [tex]\(\triangle XYZ\)[/tex] will still be congruent and similar to [tex]\(\triangle XYZ\)[/tex]. Angles remain the same.
2. Dilation by a Scale Factor of [tex]\(\frac{1}{2}\)[/tex]:
- Dilation changes the size of the figure by a specific factor while maintaining the shape. With a scale factor of [tex]\(\frac{1}{2}\)[/tex], each side of the triangle is halved, but the angles remain unchanged because dilation is a similarity transformation.
Let's evaluate each option in light of the transformations:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]:
- Both reflection and dilation are similarity transformations. Since both transformations (reflection and dilation) occur, the resulting triangles [tex]\(\triangle XYZ\)[/tex] and [tex]\(\triangle X'Y'Z'\)[/tex] are similar. Therefore, this statement is true.
2. [tex]\(\angle XZY \cong \angle Y'Z'X'\)[/tex]:
- Angles remain unchanged in both reflection and dilation. Hence, the corresponding angles [tex]\(\angle XZY\)[/tex] and [tex]\(\angle Y'Z'X'\)[/tex] must be congruent. Therefore, this statement is true.
3. [tex]\(\overline{YX} \cong \overline{Y'X'}\)[/tex]:
- Under dilation with a scale factor of [tex]\(\frac{1}{2}\)[/tex], the corresponding sides are halved. Therefore, [tex]\(\overline{YX}\)[/tex] is not congruent to [tex]\(\overline{Y'X'}\)[/tex]. Hence, this statement is false.
4. [tex]\(XZ = 2X'Z'\)[/tex]:
- After dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex], the new side lengths are half of the original side lengths. Therefore, the original side length [tex]\(XZ\)[/tex] is twice the length of the new side [tex]\(X'Z'\)[/tex]. Hence, this statement is true.
5. [tex]\(m\angle YXZ = 2 m\angle Y'X'Z'\)[/tex]:
- The measure of angles does not change under dilation. Therefore, [tex]\(m\angle YXZ\)[/tex] is equal to [tex]\(m\angle Y'X'Z'\)[/tex], not twice. Hence, this statement is false.
To summarize, three statements that must be true are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(\angle XZY \cong \angle Y'Z'X'\)[/tex]
4. [tex]\(XZ = 2X'Z'\)[/tex]
These reflect the relationships and properties preserved during the reflection and dilation transformations.
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