Certainly! Let's unpack the problem step by step.
We are given that a point, when reflected over the line [tex]\( y = -x \)[/tex], results in the coordinates [tex]\((-4, 9)\)[/tex]. Our task is to find the original coordinates of that point before the reflection, which we call the pre-image.
### Understanding Reflection Over [tex]\( y = -x \)[/tex]
When a point [tex]\((x, y)\)[/tex] is reflected over the line [tex]\( y = -x \)[/tex], its coordinates transform according to the rule:
[tex]\[ (x, y) \rightarrow (y, x) \][/tex]
However, because it's a reflection over the line [tex]\( y = -x \)[/tex], the coordinates also change their signs:
[tex]\[ (x, y) \rightarrow (-y, -x) \][/tex]
### Applying the Reflection Rule
Given the reflected coordinates are [tex]\((-4, 9)\)[/tex], we need to determine the pre-image that, when reflected over [tex]\( y = -x \)[/tex], results in these coordinates.
Using the reflection rule:
[tex]\[
(-y, -x) = (-4, 9)
\][/tex]
Let's break this down:
- The x-coordinate of the pre-image is [tex]\( -y \)[/tex] and it equals [tex]\(-4\)[/tex]. Therefore:
[tex]\[
x = 9
\][/tex]
- The y-coordinate of the pre-image is [tex]\( -x \)[/tex] and it equals [tex]\( 9 \)[/tex]. Therefore:
[tex]\[
y = -4
\][/tex]
Thus, the coordinates of the pre-image are:
[tex]\[
(x, y) = (9, -4)
\][/tex]
### Checking the Provided Options
The given options are:
1. [tex]\((-9, 4)\)[/tex]
2. [tex]\((-4, -9)\)[/tex]
3. [tex]\((4, 9)\)[/tex]
4. [tex]\( (9, -4) \)[/tex]
From our derivation, we find that the correct pre-image coordinates are:
[tex]\[
(9, -4)
\][/tex]
Thus, the correct answer is:
[tex]\[ (9, -4) \][/tex]
