[Math] Perpendicular line passing through the midpoint of another line

algebra-precalculusgeometry

I have several $2d$ line segments. for example, if I take a one line segment having end points $(x_1, y_1)$ and $(x_2, y_2)$. Then, I want to make a perpendicular line which passes through the midpoint of that line segment. for the simplicity if I say mid point $(x_3, y_3)$ then, how could I derive.

Any idea please. Thanks.

Best Answer

The midpoint of the line segment through $(x_1,y_1)$ and $(x_2,y_2)$ is $\left({x_1+x_2\over 2},{y_1+y_2\over 2}\right)$.

Also, perpendicular lines have negative reciprocal slopes, so if you want a line perpendicular to a line with slope $m$, you want to use a slope of $-{1\over m}$.

Finally, use the point-slope formula for the equation of a line through, say, $(x_3,y_3)$ with slope $m$: $y-y_3=m(x-x_3)$.

So, for example, if you want the perpendicular bisector to the segment through $P(x_1,y_1)$ and $Q(x_2,y_2)$:

  1. The midpoint is $\left({x_1+x_2\over 2},{y_1+y_2\over 2}\right)$.
  2. The slope of $\overline{PQ}$ is $m={y_2-y_1\over x_2-x_1}$, so the slope of a perpendicular is $-{1\over m}={x_1-x_2\over y_2-y_1}$.
  3. Finally, the equation of the perpendicular bisector of $\overline{PQ}$ is then $$y- {y_1+y_2\over 2}={x_1-x_2\over y_2-y_1}\left(x-{x_1+x_2\over 2}\right).$$
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