Let's evaluate the transformation steps and their effects on triangle [tex]\(ABC\)[/tex]:
1. Reflection across the [tex]\(y\)[/tex]-axis:
- When a triangle is reflected across the [tex]\(y\)[/tex]-axis, each point [tex]\((x, y)\)[/tex] of the triangle is mapped to [tex]\((-x, y)\)[/tex]. This transformation affects the triangle by flipping it to the other side of the [tex]\(y\)[/tex]-axis. Importantly, a reflection is an isometric transformation, which means it preserves the side lengths and angles of the triangle. Thus, after reflection, the size and shape of the triangle remain unchanged, although its position is mirrored.
2. Dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] centered at the origin:
- A dilation scales the triangle by a given factor relative to the center of dilation. In this case, the center is the origin [tex]\((0, 0)\)[/tex] and the scaling factor is [tex]\(\frac{1}{2}\)[/tex]. This means that every point [tex]\((x, y)\)[/tex] of the reflected triangle is mapped to [tex]\(\left(\frac{x}{2}, \frac{y}{2}\right)\)[/tex]. A dilation with a factor different from 1 does not preserve the side lengths of the triangle; instead, it scales them down by the dilation factor. However, dilation does preserve the angles of the triangle.
Now let’s summarize the transformations:
- The reflection across the [tex]\(y\)[/tex]-axis preserves both the side lengths and the angles of the triangle [tex]\(ABC\)[/tex].
- The dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] changes the side lengths but preserves the angles of the triangle [tex]\(ABC\)[/tex].
Therefore, the statement correctly describing the resulting image, triangle DEF, is:
B. Neither the reflection nor the dilation preserves the side lengths and angles of triangle [tex]\(ABC\)[/tex].
