To illustrate a perpendicular line that is not a bisector, let's consider a scenario in a 2D coordinate system.
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Choose a line: Let's represent a line with the equation \(y = 2x + 1\). This line has a slope of 2.
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Identify a point not on the line: Choose a point not on this line, say \(P(2, 5)\).
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Find the slope of the perpendicular line: The slope of a line perpendicular to a line with slope \(m\) is given by \(-\frac{1}{m}\). Since the slope of the first line is 2, the slope of the perpendicular line will be \(-\frac{1}{2}\).
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Use the point-slope form to find the perpendicular line through point \(P\): The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Substituting in \(P(2, 5)\) and the perpendicular slope \(-\frac{1}{2}\): \[ y - 5 = -\frac{1}{2}(x - 2) \] Expanding this, we have: \[ y - 5 = -\frac{1}{2}x + 1 \] \[ y = -\frac{1}{2}x + 6 \]
Now we have two lines:
- Line 1: \(y = 2x + 1\) (not bisected at any point)
- Perpendicular Line: \(y = -\frac{1}{2}x + 6\)
Conclusion: The line \(y = -\frac{1}{2}x + 6\) is perpendicular to \(y = 2x + 1\) but is not a bisector because it does not divide the original line into two equal segments at any meeting point. In fact, it intersects the original line at only one point, not bisecting it.