To find the image of point \((-4, -10)\) under the same reflection that maps point \(N(7, 8)\) to point \(N'(7, -8)\), we first need to determine the line of reflection.
The reflection of point \(N\) to point \(N'\) can be analyzed as follows.
- The x-coordinate remains the same (both are \(7\)), indicating that the line of reflection is vertical (a vertical line).
- The y-coordinates change from \(8\) to \(-8\).
This means that the line of reflection is the horizontal line that passes through the midpoint of points \(N\) and \(N'\). The coordinates for the midpoint \(M\) can be calculated as:
\[ M_x = \frac{x_1 + x_2}{2} = \frac{7 + 7}{2} = 7, \] \[ M_y = \frac{y_1 + y_2}{2} = \frac{8 + (-8)}{2} = \frac{0}{2} = 0. \]
Thus, the line of reflection is the horizontal line \(y = 0\).
To find the image of the point \((-4, -10)\) under this reflection:
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Identify the vertical distance from point \((-4, -10)\) to the line of reflection \(y = 0\). This distance is \(10\) units downward since \(-10\) is \(10\) units less than \(0\).
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To find the image point, we reflect this point across the line \(y = 0\) by moving \(10\) units upward. Therefore, the y-coordinate of the reflected point will be:
\[ -10 + 10 = 0. \]
Since the x-coordinate does not change during reflection across a horizontal line, the x-coordinate remains \(-4\).
Thus, the image of the point \((-4, -10)\) under reflection across the line \(y = 0\) is:
\[ (-4, 10). \]
The correct answer is:
A. (–4, 10).