Which point would map onto itself after a reflection across the line [tex]y=-x[/tex]?

A. [tex]\((-4, -4)\)[/tex]
B. [tex]\((-4, 0)\)[/tex]
C. [tex]\((0, -4)\)[/tex]
D. [tex]\((4, -4)\)[/tex]



Answer :

To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the transformation involved.

When a point [tex]\((a, b)\)[/tex] is reflected across the line [tex]\( y = -x \)[/tex], it is transformed to the point [tex]\((-b, -a)\)[/tex].

Let's apply this transformation to each of the given points to see which one maps onto itself:

1. For the point [tex]\((-4, -4)\)[/tex]:
[tex]\[ (-4, -4) \rightarrow (-(-4), -(-4)) = (4, 4) \][/tex]
Therefore, [tex]\((-4, -4)\)[/tex] does not map onto itself.

2. For the point [tex]\((-4, 0)\)[/tex]:
[tex]\[ (-4, 0) \rightarrow (0, 4) \][/tex]
Therefore, [tex]\((-4, 0)\)[/tex] does not map onto itself.

3. For the point [tex]\((0, -4)\)[/tex]:
[tex]\[ (0, -4) \rightarrow (4, 0) \][/tex]
Therefore, [tex]\((0, -4)\)[/tex] does not map onto itself.

4. For the point [tex]\((4, -4)\)[/tex]:
[tex]\[ (4, -4) \rightarrow (4, -4) \][/tex]
Therefore, [tex]\((4, -4)\)[/tex] maps onto itself.

Hence, the point that maps onto itself after reflecting across the line [tex]\( y = -x \)[/tex] is [tex]\((4, -4)\)[/tex].

Thus, the answer is:
[tex]\[ \boxed{4} \][/tex]