To determine which point maps onto itself after a reflection, we need to evaluate if reflecting each point across a certain line results in it being the same point. We are given the following points:
[tex]\[
(-4, -4), (-4, 0), (0, -4), (4, -4)
\][/tex]
To solve this, we need to understand that for a point to map onto itself after reflection, it must lie on the line of reflection. The most common lines of reflection are the x-axis, y-axis, the line [tex]\( y = x \)[/tex], and the line [tex]\( y = -x \)[/tex].
Let’s verify each point individually:
1. Point [tex]\((-4, -4)\)[/tex]:
- Reflecting [tex]\((-4, -4)\)[/tex] across the line [tex]\(y = x\)[/tex] results in the point [tex]\((-4, -4)\)[/tex] itself.
- Reflecting [tex]\((-4, -4)\)[/tex] across any other symmetry axis (like [tex]\( y = -x \)[/tex]) will also map it to another point [tex]\((-4, -4)\)[/tex].
Therefore, [tex]\((-4, -4)\)[/tex] remains the same after reflection across these lines.
2. Point [tex]\((-4, 0)\)[/tex]:
- Reflecting [tex]\((-4, 0)\)[/tex] across common lines of reflection will not result in [tex]\((-4, 0)\)[/tex].
3. Point [tex]\((0, -4)\)[/tex]:
- Reflecting [tex]\((0, -4)\)[/tex] across common lines of reflection will not result in [tex]\((0, -4)\)[/tex].
4. Point [tex]\((4, -4)\)[/tex]:
- Reflecting [tex]\((4, -4)\)[/tex] across common lines of reflection will not result in [tex]\((4, -4)\)[/tex].
After evaluating all the points, it is determined that:
[tex]\[
\boxed{(-4, -4)}
\][/tex]
is the only point that maps onto itself after the reflection.
