Let's look at each statement and see what must be true given the transformations involved: reflection followed by dilation.
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]:
- Dilation transformations produce similar figures because they scale all distances by the same factor while preserving angles. Thus, [tex]\(\triangle XYZ\)[/tex] is indeed similar to [tex]\(\triangle X'Y'Z'\)[/tex].
- This statement is true.
2. [tex]\(\angle XZ'Y' \cong \angle XYZ\)[/tex] (or equivalently, [tex]\(\angle XYZ \cong \angle Y'Z'X'\)[/tex]):
- Reflection and dilation do not change the measures of the interior angles of a triangle. Both transformations preserve the angles.
- Therefore, the angles in [tex]\(\triangle XYZ\)[/tex] remain congruent to the corresponding angles in [tex]\(\triangle X'Y'Z'\)[/tex].
- This statement is true.
3. [tex]\(\overline{YX} \cong \overline{Y'X'}\)[/tex]:
- Dilation changes the lengths of the sides by the scale factor. Hence, [tex]\(\overline{YX}\)[/tex] will be scaled by a factor of [tex]\(\frac{1}{2}\)[/tex]. Therefore, [tex]\(\overline{YX}\)[/tex] cannot be congruent to [tex]\(\overline{Y'X'}\)[/tex] unless [tex]\(\overline{YX}\)[/tex] were initially zero, which it is not.
- This statement is false.
4. [tex]\(XZ = 2 \cdot X'Z'\)[/tex]:
- After a dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex], the original lengths will be twice the lengths of the dilated triangle. Thus, [tex]\(XZ = 2 \cdot X'Z'\)[/tex].
- This statement is true.
5. [tex]\(m\angle YXZ = 2 \cdot m\angle Y'X'Z'\)[/tex]:
- Angle measures are invariant under dilation and reflection. Therefore, the measure of any angle in [tex]\(\triangle XY'Z'\)[/tex] will not be double the measure of the corresponding angle in [tex]\(\triangle XYZ\)[/tex].
- This statement is false.
So, the three true statements are:
1. [tex]\(\triangle XYZ \sim \triangle X'Y'Z'\)[/tex]
2. [tex]\(\angle XZ'Y' \cong \angle XYZ\)[/tex]
4. [tex]\(XZ = 2 \cdot X'Z'\)[/tex]
