To find the equations for the lines that are parallel and perpendicular to the given line passing through the point \((44, 1919)\), we need to follow these steps:
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Identify the slope of the original line:
The given line is in the slope-intercept form \(y = \frac{1}{2}x + 1111\). The slope (\(m\)) of this line is \(\frac{1}{2}\). -
Find the equation of the parallel line:
Lines that are parallel have the same slope. Therefore, the parallel line will also have a slope of \(\frac{1}{2}\). We can use point-slope form to write the equation: \[ y - y_1 = m(x - x_1) \] Plugging in the point \((44, 1919)\) and the slope \(\frac{1}{2}\): \[ y - 1919 = \frac{1}{2}(x - 44) \] Distributing: \[ y - 1919 = \frac{1}{2}x - 22 \] Adding \(1919\) to both sides gives: \[ y = \frac{1}{2}x + 1897 \] -
Find the equation of the perpendicular line:
The slope of a line that is perpendicular to another line is the negative reciprocal. The slope of the original line is \(\frac{1}{2}\), so the slope of the perpendicular line is \(-2\) (the negative reciprocal of \(\frac{1}{2}\)). Again, using the point-slope form: \[ y - 1919 = -2(x - 44) \] Distributing: \[ y - 1919 = -2x + 88 \] Adding \(1919\) to both sides gives: \[ y = -2x + 2007 \]
So, the equations of the lines are:
- Parallel: \(y = \frac{1}{2}x + 1897\)
- Perpendicular: \(y = -2x + 2007\)
None of the options given (A, B, or C) matches these results. If you need to adjust values in the options or double-check the given equations, please check again.