Let's explore the rule for reflection across the line [tex]\( y = -x \)[/tex]. When a point [tex]\( (x, y) \)[/tex] is reflected across this line, its coordinates are swapped and their signs are inverted. Specifically, the point [tex]\( (x, y) \)[/tex] becomes [tex]\( (-y, -x) \)[/tex].
Given the reflected point is [tex]\((-4, 9)\)[/tex], we need to find the original point, or pre-image, that results in this coordinate after reflection.
Let's denote the original point by [tex]\( (a, b) \)[/tex]. According to the reflection rule:
1. Swap the coordinates [tex]\( (a, b) \)[/tex] and change their signs.
2. The point [tex]\( (a, b) \)[/tex] becomes [tex]\( (-b, -a) \)[/tex].
Given:
- After reflection, the coordinates are [tex]\((-4, 9)\)[/tex].
From the reflection rule:
- The original x-coordinate [tex]\( -b \)[/tex] must equal [tex]\(-4\)[/tex].
- The original y-coordinate [tex]\( -a \)[/tex] must equal [tex]\(9\)[/tex].
To find the original coordinates:
1. Solve for [tex]\( b \)[/tex]:
[tex]\[
-b = -4 \implies b = 4
\][/tex]
2. Solve for [tex]\( a \)[/tex]:
[tex]\[
-a = 9 \implies a = -9
\][/tex]
Therefore, the original point, or the pre-image, is:
[tex]\[
(a, b) = (-9, 4)
\][/tex]
So the correct answer is:
[tex]\[
(-9, 4)
\][/tex]
