Answer :
To determine which statements about the triangles [tex]\( \triangle XYZ \)[/tex] and [tex]\( \triangle X'Y'Z' \)[/tex] must be true following a reflection over a vertical line and a dilation by a scale factor of [tex]\( \frac{1}{2} \)[/tex], let's analyze each given option step-by-step:
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]:
Similarity between triangles means that their corresponding angles are equal and the lengths of corresponding sides are proportional.
- Reflection over a vertical line does not change the shape or angles of the triangle.
- Dilation by a scale factor only changes the size but maintains the proportions.
Therefore, the triangles remain similar.
This statement is true.
2. [tex]\( \angle XZY \cong \angle YZ'X' \)[/tex]:
Congruence of angles means that they are equal in measure.
- Reflecting over a vertical line does not change the angle measures.
- Dilation does not change angle measures either.
Thus, the angle [tex]\( \angle XZY \)[/tex] must be congruent to the corresponding angle in [tex]\( \triangle X'Y'Z' \)[/tex], which is [tex]\( \angle YZ'X' \)[/tex].
This statement is true.
3. [tex]\( \overline{YX} \cong \overline{Y'X} \)[/tex]:
Congruence of segments means that they are equal in length.
- Reflecting a shape over a line and then dilating it by a factor [tex]\( \frac{1}{2} \)[/tex] reduces each side length by half.
Therefore, the segments [tex]\( YX \)[/tex] from [tex]\( \triangle XYZ \)[/tex] and [tex]\( Y'X \)[/tex] from [tex]\( \triangle X'Y'Z' \)[/tex] cannot be equal in length because [tex]\( Y'X \)[/tex] is half as long as [tex]\( YX \)[/tex].
This statement is false.
4. [tex]\( XZ = 2X'Z' \)[/tex]:
This compares the lengths of the same side before and after dilation.
- Dilation by a scale factor of [tex]\( \frac{1}{2} \)[/tex] implies that any side of the triangle in [tex]\( \triangle XYZ \)[/tex] is twice as long as the corresponding side in [tex]\( \triangle X'Y'Z' \)[/tex].
Thus, [tex]\( XZ \)[/tex] must be twice the length of [tex]\( X'Z' \)[/tex].
This statement is true.
5. [tex]\( m \angle YXZ = 2m \angle YX'Z' \)[/tex]:
This compares the measures of corresponding angles.
- As established, dilation affects side lengths but not angle measures.
Therefore, the angle measure remains the same and is not scaled by the dilation factor.
This statement is false.
So, the three statements that must be true are:
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex].
2. [tex]\( \angle XZY \cong \angle YZ'X' \)[/tex].
4. [tex]\( XZ = 2X'Z' \)[/tex].
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex]:
Similarity between triangles means that their corresponding angles are equal and the lengths of corresponding sides are proportional.
- Reflection over a vertical line does not change the shape or angles of the triangle.
- Dilation by a scale factor only changes the size but maintains the proportions.
Therefore, the triangles remain similar.
This statement is true.
2. [tex]\( \angle XZY \cong \angle YZ'X' \)[/tex]:
Congruence of angles means that they are equal in measure.
- Reflecting over a vertical line does not change the angle measures.
- Dilation does not change angle measures either.
Thus, the angle [tex]\( \angle XZY \)[/tex] must be congruent to the corresponding angle in [tex]\( \triangle X'Y'Z' \)[/tex], which is [tex]\( \angle YZ'X' \)[/tex].
This statement is true.
3. [tex]\( \overline{YX} \cong \overline{Y'X} \)[/tex]:
Congruence of segments means that they are equal in length.
- Reflecting a shape over a line and then dilating it by a factor [tex]\( \frac{1}{2} \)[/tex] reduces each side length by half.
Therefore, the segments [tex]\( YX \)[/tex] from [tex]\( \triangle XYZ \)[/tex] and [tex]\( Y'X \)[/tex] from [tex]\( \triangle X'Y'Z' \)[/tex] cannot be equal in length because [tex]\( Y'X \)[/tex] is half as long as [tex]\( YX \)[/tex].
This statement is false.
4. [tex]\( XZ = 2X'Z' \)[/tex]:
This compares the lengths of the same side before and after dilation.
- Dilation by a scale factor of [tex]\( \frac{1}{2} \)[/tex] implies that any side of the triangle in [tex]\( \triangle XYZ \)[/tex] is twice as long as the corresponding side in [tex]\( \triangle X'Y'Z' \)[/tex].
Thus, [tex]\( XZ \)[/tex] must be twice the length of [tex]\( X'Z' \)[/tex].
This statement is true.
5. [tex]\( m \angle YXZ = 2m \angle YX'Z' \)[/tex]:
This compares the measures of corresponding angles.
- As established, dilation affects side lengths but not angle measures.
Therefore, the angle measure remains the same and is not scaled by the dilation factor.
This statement is false.
So, the three statements that must be true are:
1. [tex]\( \triangle XYZ \sim \triangle X'Y'Z' \)[/tex].
2. [tex]\( \angle XZY \cong \angle YZ'X' \)[/tex].
4. [tex]\( XZ = 2X'Z' \)[/tex].
