To determine the correct description of the resulting image, triangle DEF, after a reflection across the [tex]\(y\)[/tex]-axis and a dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] centered at the origin, we need to understand the properties of reflections and dilations.
Firstly, let's consider the properties of reflection across the [tex]\(y\)[/tex]-axis:
- A reflection is a type of rigid transformation.
- It preserves the side lengths and angles of the original figure.
- Each point of the original figure is mapped to a point directly on the opposite side of the [tex]\(y\)[/tex]-axis at an equal distance.
Next, let's examine the properties of dilation with a center at the origin and a factor of [tex]\(\frac{1}{2}\)[/tex]:
- A dilation is a non-rigid transformation.
- It changes the size of the original figure while preserving the shape.
- Specifically, it reduces the side lengths of the original figure by a factor of [tex]\(\frac{1}{2}\)[/tex], but it does not alter the angles.
Now, let's apply these properties to triangle ABC:
1. Reflection across the [tex]\(y\)[/tex]-axis:
- The reflected triangle, let's call it triangle [tex]\( \text{A'} \text{B'} \text{C'} \)[/tex], will have the same side lengths and angles as triangle ABC.
2. Dilation by a factor of [tex]\(\frac{1}{2}\)[/tex]:
- The dilated triangle, let's denote it as triangle DEF, will have sides half the length of the sides of triangle [tex]\( \text{A'} \text{B'} \text{C'} \)[/tex], but all the angles will remain the same as those in triangle [tex]\( \text{A'} \text{B'} \text{C'} \)[/tex] and, consequently, triangle ABC.
Thus, considering the nature of these transformations, the correct state describing the resulting triangle DEF is:
- The reflection preserves the side lengths and angles of triangle ABC. The dilation preserves angles but not side lengths.
Therefore, the correct answer is:
[tex]\[ \boxed{2 \text{ (B)} } \][/tex]
The reflection and dilation together lead us to this conclusion.
