Let's break down the transformations step by step and see which properties they preserve in a triangle.
### Step 1: Reflection Across the [tex]$y$[/tex]-Axis
When triangle [tex]$ABC$[/tex] is reflected across the [tex]$y$[/tex]-axis, each point of the triangle [tex]$(x, y)$[/tex] is transformed to [tex]$(-x, y)$[/tex]. This transformation has the following properties:
- Side Lengths: Reflection preserves the side lengths of the triangle because the distance between points remains the same.
- Angles: Reflection also preserves the angles within the triangle since the shape is only flipped and not deformed.
Thus, the reflection across the [tex]$y$[/tex]-axis preserves both the side lengths and angles of triangle [tex]$ABC$[/tex].
### Step 2: Dilation by a Factor of [tex]$\frac{1}{2}$[/tex] Centered at the Origin
Dilation involves scaling the triangle by a certain factor relative to a center point (in this case, the origin). When dilating by a factor of [tex]$\frac{1}{2}$[/tex]:
- Side Lengths: The side lengths of the triangle are not preserved. Each side length of the triangle is scaled by the factor of [tex]$\frac{1}{2}$[/tex], which means that every side length in the resulting triangle will be half of its original length.
- Angles: The angles within the triangle are preserved during dilation. This is because scaling alters distances but not the relative directional slopes between the sides, which determine the angles.
Thus, dilation by a factor of [tex]$\frac{1}{2}$[/tex] preserves the angles but not the side lengths of triangle [tex]$ABC$[/tex].
### Correct Description
From the above analysis:
- Reflection preserves both side lengths and angles.
- Dilation preserves angles but not side lengths.
Therefore, the statement that correctly describes the resulting image, triangle DEF, is:
C. The reflection preserves the side lengths and angles of triangle [tex]$ABC$[/tex]. The dilation preserves angles but not side lengths.
