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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]

Sagot :

GinnyAnswer
To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand how reflection works in this context. When a point [tex]\((x, y)\)[/tex] is reflected across the line [tex]\( y = -x \)[/tex], its coordinates become [tex]\((-y, -x)\)[/tex].

Let's reflect each given point and see which one maps onto itself.

1. Point [tex]\((-4, -4)\)[/tex]:
- Reflect [tex]\((-4, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex].
- The new coordinates would be [tex]\((-(-4), -(-4)) = (4, 4)\)[/tex].
- [tex]\((-4, -4) \neq (4, 4)\)[/tex].

2. Point [tex]\((-4, 0)\)[/tex]:
- Reflect [tex]\((-4, 0)\)[/tex] across the line [tex]\( y = -x \)[/tex].
- The new coordinates would be [tex]\((0, 4)\)[/tex].
- [tex]\((-4, 0) \neq (0, 4)\)[/tex].

3. Point [tex]\((0, -4)\)[/tex]:
- Reflect [tex]\((0, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex].
- The new coordinates would be [tex]\((4, 0)\)[/tex].
- [tex]\((0, -4) \neq (4, 0)\)[/tex].

4. Point [tex]\((4, -4)\)[/tex]:
- Reflect [tex]\((4, -4)\)[/tex] across the line [tex]\( y = -x \)[/tex].
- The new coordinates would be [tex]\((4, -4)\)[/tex].
- [tex]\((4, -4) = (4, -4)\)[/tex].

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

Therefore, the answer is the fourth point: [tex]\((4, -4)\)[/tex].
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