Let's analyze the transformations given in the problem and understand their effects on the properties of triangle ABC.
1. Reflection across the y-axis:
- When a triangle is reflected across the y-axis, its shape and size remain unchanged. The reflection preserves both the side lengths and the angles of the triangle. Essentially, the coordinates of any point [tex]\((x, y)\)[/tex] are transformed to [tex]\((-x, y)\)[/tex]. This does not alter the distances between points or the angles.
Therefore, the reflection preserves the side lengths and angles of triangle ABC.
2. Dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] centered at the origin:
- Dilation is a transformation that stretches or shrinks a figure by multiplying the distance of each point from the center of dilation by a scale factor (in this case, [tex]\(\frac{1}{2}\)[/tex]). Dilation preserves the angles but changes the side lengths in proportion to the scale factor.
- Since the dilation factor is [tex]\(\frac{1}{2}\)[/tex], every side length of the triangle ABC will be halved, but the angles will remain the same.
Therefore, dilation preserves the angles but not the side lengths of triangle ABC.
Based on this analysis, the correct statement describing the resulting image, triangle DEF, after the reflection and dilation, is:
- The reflection preserves the side lengths and angles of triangle ABC. The dilation preserves angles but not side lengths.
Thus, the correct answer is:
C. The reflection preserves the side lengths and angles of triangle [tex]$A B C$[/tex]. The dilation preserves angles but not side lengths.
