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

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



Answer :

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

A point [tex]\((x, y)\)[/tex] will reflect to a point [tex]\((-y, -x)\)[/tex] when reflected across the line [tex]\( y = -x \)[/tex]. Essentially, we swap the coordinates and change their signs.

Now, we need to determine which point remains unchanged under this transformation. This means we need to solve for the point [tex]\((x, y)\)[/tex] such that:
[tex]\[ (x, y) = (-y, -x) \][/tex]

Let's test each given point:

1. For [tex]\((-4, -4)\)[/tex]:
[tex]\[ (x, y) = (-4, -4) \][/tex]
[tex]\[ (-y, -x) = (--4, --4) = (4, 4) \][/tex]
After reflection: [tex]\((4, 4)\)[/tex]. This does not map onto itself.

2. For [tex]\((-4, 0)\)[/tex]:
[tex]\[ (x, y) = (-4, 0) \][/tex]
[tex]\[ (-y, -x) = (0, 4) \][/tex]
After reflection: [tex]\((0, 4)\)[/tex]. This does not map onto itself.

3. For [tex]\((0, -4)\)[/tex]:
[tex]\[ (x, y) = (0, -4) \][/tex]
[tex]\[ (-y, -x) = (4, 0) \][/tex]
After reflection: [tex]\((4, 0)\)[/tex]. This does not map onto itself.

4. For [tex]\((4, -4)\)[/tex]:
[tex]\[ (x, y) = (4, -4) \][/tex]
[tex]\[ (-y, -x) = (4, -4) \][/tex]
After reflection: [tex]\((4, -4)\)[/tex]. This point maps onto itself.

Therefore, the correct point that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex] is:
[tex]\[ (4, -4) \][/tex]

So, the point that would map onto itself is the fourth point:
[tex]\[ (4, -4) \][/tex]