To determine the correct statement that describes the properties of triangles [tex]\(XYZ\)[/tex] and [tex]\(XYZ''\)[/tex] after the translation and dilation transformations, let’s analyze each option carefully.
### Translation
1. Rule [tex]$(x+1, y-1)$[/tex]:
- This is a translation which shifts every point of the triangle 1 unit to the right and 1 unit down.
Let's examine how this affects the properties:
- Angles: The angles of the triangle remain the same because translation is a rigid motion.
- Side lengths: The lengths of the sides of the triangle also remain the same for the same reason – a rigid motion (translation), does not alter lengths or angles.
### Dilation
2. Scale factor of 4 centered at the origin:
- This is a dilation which scales the positions of all points of the triangle by multiplying their distances from the origin by 4.
Let's examine how this affects the properties:
- Angles: The angles of the triangle remain the same since dilation maintains the shape but changes the size. Angles are not altered by scaling.
- Side lengths: The lengths of the sides of the triangle change by a proportion equal to the scale factor (here, multiplied by 4).
### Analyzing the Options
1. [tex]$\angle Y$[/tex] and [tex]$\angle Y^$[/tex] are congruent after the translation, but not after the dilation.
- Incorrect because angles remain the same (congruent) during both translation and dilation.
2. [tex]$\angle Y$[/tex] and [tex]$\angle Y^$[/tex] are congruent after the dilation, but not after the translation.
- Incorrect because angles are congruent during both translation and dilation.
3. [tex]$\overline{Y Z}$[/tex] and [tex]$\overline{Y^{\prime \prime} Z^{\prime \prime}}$[/tex] are proportional after the translation, but not after the dilation.
- Incorrect because translation does not change the lengths (they remain congruent). After dilation, lengths are in proportion by the scale factor.
4. [tex]$\overline{Y Z}$[/tex] and [tex]$\overline{Y^{\prime \prime} Z^{\prime \prime}}$[/tex] are proportional after the dilation and congruent after the translation.
- This is correct since:
- After translation, side lengths remain unchanged, so they are congruent.
- After dilation, side lengths change by a constant factor (scale factor of 4), making the segments proportional.
### Conclusion
The correct statement is:
[tex]$\overline{YZ}$[/tex] and [tex]$\overline{Y^{\prime \prime} Z^{\prime \prime}}$[/tex] are proportional after the dilation and congruent after the translation.
