[tex]$\triangle XYZ$[/tex] was reflected over a vertical line, then dilated by a scale factor of [tex]$\frac{1}{2}$[/tex], resulting in [tex]$\triangle X^{\prime}Y^{\prime}Z^{\prime}$[/tex]. Which must be true of the two triangles? Select three options.

A. [tex]$\triangle XYZ \sim \triangle X^{\prime}Y^{\prime}Z^{\prime}$[/tex]

B. [tex]$\angle XZY \cong \angle Y^{\prime}Z^{\prime}X^{\prime}$[/tex]

C. [tex]$\overline{YX} \cong \overline{Y^{\prime}X^{\prime}}$[/tex]

D. [tex]$XZ = 2X^{\prime}Z^{\prime}$[/tex]

E. [tex]$m\angle YXZ = 2m\angle Y^{\prime}X^{\prime}Z^{\prime}$[/tex]



Answer :

To determine which statements must be true for the given transformations, let's consider the properties of reflection and dilation.

1. Similarity of triangles:
- Reflections and dilations both preserve the shape of a triangle, retaining its angles.
- After reflecting [tex]$\triangle XYZ$[/tex], the new triangle is congruent to the original in terms of angle measures but might be positioned differently.
- Dilation with a scale factor of [tex]$\frac{1}{2}$[/tex] changes the size of the triangle but keeps the angles the same and side lengths proportional.
- Therefore, [tex]$\triangle XYZ$[/tex] and [tex]$\triangle X'Y'Z'$[/tex] are similar triangles because their corresponding angles are equal and the side lengths are proportional.

[tex]\[ \boxed{\triangle XYZ \sim \triangle X'Y'Z'} \][/tex]

2. Angle congruence:
- Reflecting a triangle over a vertical line does not change the measures of the angles; it only changes the orientation.
- Thus, the angles of [tex]$\triangle XYZ$[/tex] are congruent to the corresponding angles of [tex]$\triangle X'Y'Z'$[/tex].
- Specifically, [tex]$\angle XZY$[/tex] from [tex]$\triangle XYZ$[/tex] will be congruent to [tex]$\angle Y'Z'X'$[/tex] from [tex]$\triangle X'Y'Z'$[/tex] after reflection.

[tex]\[ \boxed{\angle XZY \cong \angle Y'Z'X'} \][/tex]

3. Congruence of sides:
- After reflecting [tex]$\triangle XYZ$[/tex], the lengths of the sides do not change.
- Dilation with a scale factor of [tex]$\frac{1}{2}$[/tex] means the new triangle's sides are half the length of the original triangle’s sides.
- Therefore, the side lengths in [tex]$\triangle X'Y'Z'$[/tex] are not congruent to the original triangle's sides but are scaled down by a factor of [tex]$\frac{1}{2}$[/tex].

[tex]\[ \not\cong \boxed{\overline{YX} \not\cong \overline{Y'X'}} \][/tex]

4. Proportionality of sides:
- If [tex]$XZ$[/tex] is a side in the original triangle [tex]$\triangle XYZ$[/tex] and [tex]$X'Z'$[/tex] is the corresponding side in the dilated triangle, then the length of [tex]$XZ$[/tex] will be twice the length of [tex]$X'Z'$[/tex] because the dilation factor is [tex]$\frac{1}{2}$[/tex].
- Thus, [tex]$XZ = 2 \cdot X'Z'$[/tex].

[tex]\[ \boxed{XZ = 2 \cdot X'Z'} \][/tex]

5. Angle measure doubling:
- Dilation changes the side lengths of the triangle but does not alter the measures of the angles.
- Therefore, [tex]$m \angle YXZ$[/tex] is equal to [tex]$m \angle Y'X'Z'$[/tex], and it is not doubled.

[tex]\[ \not\boxed{m \angle YXZ \not= 2 \cdot m \angle Y'X'Z'} \][/tex]

The three correct statements that must be true based on the transformations of [tex]$\triangle XYZ$[/tex] to [tex]$\triangle X'Y'Z'$[/tex] are:

1. [tex]$\triangle XYZ \sim \triangle X'Y'Z'$[/tex].
2. [tex]$\angle XZY \cong \angle Y'Z'X'$[/tex].
4. [tex]$XZ = 2 \cdot X'Z'$[/tex].