Answer :
To solve this problem, we need to analyze the effects of both a reflection across the [tex]$y$[/tex]-axis and a dilation centered at the origin by a factor of [tex]$\frac{1}{2}$[/tex] on triangle [tex]$ABC$[/tex].
1. Reflection Across the [tex]$y$[/tex]-Axis:
- A reflection across the [tex]$y$[/tex]-axis transforms a point [tex]$(x, y)$[/tex] to [tex]$(-x, y)$[/tex].
- This transformation is a rigid motion, meaning it preserves the distances (side lengths) and angles of the figure.
- Therefore, when triangle [tex]$ABC$[/tex] is reflected across the [tex]$y$[/tex]-axis, the resulting triangle will have the same side lengths and angles as the original triangle [tex]$ABC$[/tex].
2. Dilation by a Factor of [tex]$\frac{1}{2}$[/tex] Centered at the Origin:
- A dilation centered at the origin by a factor of [tex]$\frac{1}{2}$[/tex] transforms a point [tex]$(x, y)$[/tex] to [tex]$(\frac{x}{2}, \frac{y}{2})$[/tex].
- Dilations centered at a point scale the distances from that point by the given factor. In this case, every point will move closer to the origin by half the original distance.
- Dilations preserve the angles of the figure but not the side lengths. The side lengths will be half of what they were in the original figure.
Putting it all together:
- The reflection will result in a triangle congruent to [tex]$ABC$[/tex] (same side lengths and angles).
- The subsequent dilation will preserve the angles of the reflected triangle but will reduce all side lengths to half.
So, the correct statement is:
A. The reflection preserves the side lengths and angles of triangle [tex]$ABC$[/tex]. The dilation preserves angles but not side lengths.
1. Reflection Across the [tex]$y$[/tex]-Axis:
- A reflection across the [tex]$y$[/tex]-axis transforms a point [tex]$(x, y)$[/tex] to [tex]$(-x, y)$[/tex].
- This transformation is a rigid motion, meaning it preserves the distances (side lengths) and angles of the figure.
- Therefore, when triangle [tex]$ABC$[/tex] is reflected across the [tex]$y$[/tex]-axis, the resulting triangle will have the same side lengths and angles as the original triangle [tex]$ABC$[/tex].
2. Dilation by a Factor of [tex]$\frac{1}{2}$[/tex] Centered at the Origin:
- A dilation centered at the origin by a factor of [tex]$\frac{1}{2}$[/tex] transforms a point [tex]$(x, y)$[/tex] to [tex]$(\frac{x}{2}, \frac{y}{2})$[/tex].
- Dilations centered at a point scale the distances from that point by the given factor. In this case, every point will move closer to the origin by half the original distance.
- Dilations preserve the angles of the figure but not the side lengths. The side lengths will be half of what they were in the original figure.
Putting it all together:
- The reflection will result in a triangle congruent to [tex]$ABC$[/tex] (same side lengths and angles).
- The subsequent dilation will preserve the angles of the reflected triangle but will reduce all side lengths to half.
So, the correct statement is:
A. The reflection preserves the side lengths and angles of triangle [tex]$ABC$[/tex]. The dilation preserves angles but not side lengths.
