To find the coordinates of vertex X' after reflecting vertex X over a horizontal line, we first need to determine the horizontal line of reflection.
Given the original coordinates of vertex X, which are (-3, 9), we can use point X' (3, -2) to find the y-coordinate of the line of reflection.
When reflecting a point over a horizontal line, the y-coordinate of the point and its reflection have the same distance to the line.
Let \( h \) be the y-coordinate of the line of reflection. The distance from X to the line is given by:
\[ d = 9 - h \]
And for X', we have:
\[ d' = h - (-2) = h + 2 \]
Setting these two distances equal to each other gives:
\[ 9 - h = h + 2 \]
Rearranging this equation:
\[ 9 - 2 = 2h \] \[ 7 = 2h \] \[ h = \frac{7}{2} = 3.5 \]
Now, using the line of reflection at \( y = 3.5 \), we can find the coordinates of the reflected point X'.
The y-coordinate of the point X (9) and the distance to the line (3.5) will have an equal distance from the line going downward to get the reflected point's y-coordinate (-2):
\[ 3.5 - 9 = -5.5 \] \[ 3.5 - (-2) = 5.5 \]
The reflection went down 5.5 units to turn 9 to -2:
Thus the y-coordinate of vertex X is:
\[ y = 3.5 + (3.5 - 9) = 3.5 - 5.5 = -2 \]
Keeping the x-coordinate the same, we find the coordinates of vertex X after reflection over the line of 3.5 is:
The reflection stays at \( x = -3 \).
Calculating the original coordinates, we have:
Thus, the coordinates of vertex X before reflection is:
\[ (-3, 9) \]
But to find the vertex listed in the options, we just need to assess the choices. In the problem, we need to give the options; with our correct understanding of the reflected options and the remaining vertices, there might be confusion.
The answer isn't listed as it should be vertex representations needing clarity, leading to options being:
Since no vertex matches the coordinate options correctly. None of the answers exactly correspond to (-3, 9).
Thus, none of the coordinate options were accurately reflective across the horizontal line calculation.
It looks like the provided answers do not accurately represent the expected answer for vertex \( X \). Without correct vertex coordinates, reanalyses of the answers for comparative checks should be processed, but our vertex remains overhead unlisted.
So, ultimately, reconsideration about the problem setup likely is needed.
Please finalize which part of the reflection answer checks directly confront or seek omitted factual accounts.