To determine the line of reflection that maps triangle [tex]\(DEF\)[/tex] to its image [tex]\(D'E'F'\)[/tex] in the context of point [tex]\(E\)[/tex] being reflected to [tex]\(E'\)[/tex], we need to carefully analyze the given points and their transformations.
Given:
- Original triangle [tex]\(DEF\)[/tex] has vertices [tex]\(D(-3, 5)\)[/tex], [tex]\(E(-10, 4)\)[/tex], and [tex]\(F(-11, 8)\)[/tex].
- After reflection, point [tex]\(E\)[/tex] is mapped to [tex]\(E'(-10, -4)\)[/tex].
To find the line of reflection:
1. Reflection Property: When a point is reflected through a line, the line acts as the perpendicular bisector of the segment joining the point and its image. This means that the perpendicular distance from the original point to the line of reflection is equal to the distance from the reflected point to the line of reflection.
2. Analyze the Original and Transformed Points: We know [tex]\(E(-10,4)\)[/tex] and [tex]\(E'(-10,-4)\)[/tex].
- Both points [tex]\(E\)[/tex] and [tex]\(E'\)[/tex] share the same x-coordinate ([tex]\(x = -10\)[/tex]).
- However, the y-coordinates [tex]\(4\)[/tex] and [tex]\(-4\)[/tex] are symmetric about a central line.
3. Identify Symmetry: Since the points [tex]\(E\)[/tex] and [tex]\(E'\)[/tex] are aligned vertically and their y-coordinates are equidistant from 0, the line of reflection must be the horizontal axis (x-axis) where:
[tex]\[
y = 0
\][/tex]
To summarize, the reflection that maps [tex]\(E(-10, 4)\)[/tex] to [tex]\(E'(-10, -4)\)[/tex] happens over the [tex]\(x\)[/tex]-axis because the y-coordinates differ only in their signs, being equidistant from the [tex]\(x\)[/tex]-axis.
Hence, the line of reflection is:
[tex]\[
\boxed{x\text{-axis}}
\][/tex]
