When a point is reflected over a horizontal line, the x-coordinate remains the same while the y-coordinate changes. Specifically, if a point has coordinates \((x, y)\) and is reflected over a horizontal line that is \(k\) units below or above the original y-coordinate, the new coordinates will be \((x, 2k - y)\) where \(k\) is the y-coordinate of the line of reflection.
In this case, we have vertex \(X\) with coordinates \((-3, 9)\).
To find the coordinates of \(X'\), we need to identify the horizontal line of reflection. The only given points that can help us determine the line of reflection are points \(Z\) and \(Z'\). Point \(Z\) is at \((3, 4)\) and its reflection \(Z'\) is at \((3, -2)\).
The horizontal line of reflection can be found by taking the average of the y-coordinates of \(Z\) and \(Z'\):
\[ \text{Average y-coordinate} = \frac{4 + (-2)}{2} = \frac{2}{2} = 1 \]
Now, the line of reflection is \(y = 1\).
Next, we find \(X'\) by using the reflection formula:
The y-coordinate of point \(X\) is \(9\). To reflect \(X\) over the line \(y = 1\), we calculate the distance from \(9\) to \(1\) (which is \(8\) units above) and then subtract this distance from \(1\):
\[ \text{New y-coordinate} = 1 - (9 - 1) = 1 - 8 = -7 \]
Thus, the coordinates of \(X'\) are:
\[ X' = (-3, -7) \]
Therefore, the correct answer is:
A. (-3, -7)
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