To determine the correct statement about triangle DEF resulting from reflecting triangle ABC across the y-axis and then dilating it by a factor of [tex]\(\frac{1}{2}\)[/tex] centered at the origin, let's analyze the transformations step by step:
1. Reflection Across the y-axis:
When triangle ABC is reflected across the y-axis, each point of the triangle moves to a point directly opposite on the other side of the y-axis, while staying at the same height.
- This reflection does not change the lengths of the sides of the triangle nor the measures of the angles within the triangle. Therefore, it preserves both side lengths and angles.
2. Dilation Centered at the Origin by a Factor of [tex]\(\frac{1}{2}\)[/tex]:
Next, a dilation centered at the origin with a factor of [tex]\(\frac{1}{2}\)[/tex] scales down the size of the triangle to half its original side lengths.
- This dilation preserves the angles of the triangle but scales the side lengths proportionally by the factor [tex]\(\frac{1}{2}\)[/tex]. Thus, the side lengths are reduced by half.
Combining these two observations:
- The reflection preserves both the side lengths and angles.
- The dilation preserves the angles but reduces the side lengths of the triangle to half.
Given these facts, the statement that accurately describes the resulting image, triangle DEF, is:
B. The reflection preserves the side lengths and angles of triangle ABC. The dilation preserves angles but not side lengths.
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
