Answer :
To determine which statement correctly describes the resulting image, triangle DEF, after triangle ABC is reflected across the y-axis and then dilated by a factor of [tex]\(\frac{1}{2}\)[/tex] centered at the origin, let’s analyze the effects of both transformations individually and then in combination.
1. Reflection across the y-axis:
- A reflection across the y-axis will transform every point [tex]\((x, y)\)[/tex] of triangle ABC to [tex]\((-x, y)\)[/tex].
- This transformation preserves the shape and size of the triangle. This means the side lengths and angles of triangle ABC are maintained in the reflected image. There is no change in the lengths of the sides or the measures of the angles.
2. Dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] centered at the origin:
- Dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] will transform every point [tex]\((x, y)\)[/tex] of the reflected triangle to [tex]\(\left(\frac{x}{2}, \frac{y}{2}\right)\)[/tex].
- Dilation preserves the angles of the triangle but scales the side lengths by the factor of [tex]\(\frac{1}{2}\)[/tex]. Therefore, the sides of the resulting image will be half the length of the sides of the reflected triangle.
Now, combining these transformations:
- The reflection step: Preserves both the side lengths and the angles of the triangle.
- The dilation step: Alters the side lengths (reducing them by half) but preserves the angles of the triangle.
Therefore, the statement that correctly describes the resulting image, triangle DEF, is:
D. The reflection preserves the side lengths and angles of triangle ABC. The dilation preserves angles but not side lengths.
1. Reflection across the y-axis:
- A reflection across the y-axis will transform every point [tex]\((x, y)\)[/tex] of triangle ABC to [tex]\((-x, y)\)[/tex].
- This transformation preserves the shape and size of the triangle. This means the side lengths and angles of triangle ABC are maintained in the reflected image. There is no change in the lengths of the sides or the measures of the angles.
2. Dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] centered at the origin:
- Dilation by a factor of [tex]\(\frac{1}{2}\)[/tex] will transform every point [tex]\((x, y)\)[/tex] of the reflected triangle to [tex]\(\left(\frac{x}{2}, \frac{y}{2}\right)\)[/tex].
- Dilation preserves the angles of the triangle but scales the side lengths by the factor of [tex]\(\frac{1}{2}\)[/tex]. Therefore, the sides of the resulting image will be half the length of the sides of the reflected triangle.
Now, combining these transformations:
- The reflection step: Preserves both the side lengths and the angles of the triangle.
- The dilation step: Alters the side lengths (reducing them by half) but preserves the angles of the triangle.
Therefore, the statement that correctly describes the resulting image, triangle DEF, is:
D. The reflection preserves the side lengths and angles of triangle ABC. The dilation preserves angles but not side lengths.
