Answer :
To solve this problem, let's consider the properties of reflections and dilations separately:
### Reflection Across the [tex]\( y \)[/tex]-Axis
1. Preservation of Angles and Side Lengths:
- Angles: When a triangle is reflected across the [tex]\( y \)[/tex]-axis, the angles remain the same. This is because reflections are isometric transformations, meaning they preserve the shape and size of geometric figures, though they may change the figure's orientation.
- Side Lengths: Similar to angles, the side lengths of the triangle are also preserved because a reflection is an isometric transformation which does not alter the dimensions of the figure.
In conclusion, reflection across the [tex]\( y \)[/tex]-axis does preserve both the side lengths and the angles of the triangle.
### Dilation by a Factor of [tex]\( \frac{1}{2} \)[/tex]
1. Preservation of Angles and Side Lengths:
- Angles: Dilation is a similarity transformation that preserves the angles of the figure being transformed. Therefore, the angles of triangle [tex]\( ABC \)[/tex] will remain the same after dilating.
- Side Lengths: However, dilation alters the side lengths. Specifically, a dilation by a factor of [tex]\( \frac{1}{2} \)[/tex] will scale down all side lengths by half. Thus, the side lengths of the triangle will not be preserved.
In conclusion, dilation by a factor of [tex]\( \frac{1}{2} \)[/tex] preserves the angles but does not preserve the side lengths.
### Reviewing the Statements
Now let's review which statement correctly describes the resulting image [tex]\( \triangle DEF \)[/tex]:
A. Neither the reflection nor the dilation preserves the side lengths and angles of triangle [tex]\( ABC \)[/tex].
- This is incorrect because the reflection preserves both side lengths and angles.
B. The dilation preserves the side lengths and angles of triangle [tex]\( ABC \)[/tex]. The reflection does not preserve side lengths and angles.
- This is incorrect because the dilation does not preserve side lengths, and the reflection preserves both.
C. The reflection preserves the side lengths and angles of triangle [tex]\( ABC \)[/tex]. The dilation preserves angles but not side lengths.
- This is correct. The reflection maintains both side lengths and angles, while the dilation keeps the angles unchanged but alters the side lengths.
D. Both the reflection and dilation preserve the side lengths and angles of triangle [tex]\( ABC \)[/tex].
- This is incorrect because the dilation does not preserve the side lengths.
Based on the analysis, the correct statement is:
[tex]\[ \boxed{3 \text{ (C)}} \][/tex]
### Reflection Across the [tex]\( y \)[/tex]-Axis
1. Preservation of Angles and Side Lengths:
- Angles: When a triangle is reflected across the [tex]\( y \)[/tex]-axis, the angles remain the same. This is because reflections are isometric transformations, meaning they preserve the shape and size of geometric figures, though they may change the figure's orientation.
- Side Lengths: Similar to angles, the side lengths of the triangle are also preserved because a reflection is an isometric transformation which does not alter the dimensions of the figure.
In conclusion, reflection across the [tex]\( y \)[/tex]-axis does preserve both the side lengths and the angles of the triangle.
### Dilation by a Factor of [tex]\( \frac{1}{2} \)[/tex]
1. Preservation of Angles and Side Lengths:
- Angles: Dilation is a similarity transformation that preserves the angles of the figure being transformed. Therefore, the angles of triangle [tex]\( ABC \)[/tex] will remain the same after dilating.
- Side Lengths: However, dilation alters the side lengths. Specifically, a dilation by a factor of [tex]\( \frac{1}{2} \)[/tex] will scale down all side lengths by half. Thus, the side lengths of the triangle will not be preserved.
In conclusion, dilation by a factor of [tex]\( \frac{1}{2} \)[/tex] preserves the angles but does not preserve the side lengths.
### Reviewing the Statements
Now let's review which statement correctly describes the resulting image [tex]\( \triangle DEF \)[/tex]:
A. Neither the reflection nor the dilation preserves the side lengths and angles of triangle [tex]\( ABC \)[/tex].
- This is incorrect because the reflection preserves both side lengths and angles.
B. The dilation preserves the side lengths and angles of triangle [tex]\( ABC \)[/tex]. The reflection does not preserve side lengths and angles.
- This is incorrect because the dilation does not preserve side lengths, and the reflection preserves both.
C. The reflection preserves the side lengths and angles of triangle [tex]\( ABC \)[/tex]. The dilation preserves angles but not side lengths.
- This is correct. The reflection maintains both side lengths and angles, while the dilation keeps the angles unchanged but alters the side lengths.
D. Both the reflection and dilation preserve the side lengths and angles of triangle [tex]\( ABC \)[/tex].
- This is incorrect because the dilation does not preserve the side lengths.
Based on the analysis, the correct statement is:
[tex]\[ \boxed{3 \text{ (C)}} \][/tex]
