Question

Can you show me how to find the equation of the perpendicular bisector of the segment with endpoints (3,4) and (1,0)?

Answer 1

\[\frac{ 3+1 }{ 2 }=2\]Considering this was written an hour ago, you've probably got the answer already.

A perpendicular line is a line that has a slope that is the opposite reciprocal of the line it bisects, therefore we must find the slope of the segment, along with it's midpoint.
We will start with the slope, using point slope form. \[\frac{ x2 - x1 }{ y2-y1 }\]
So it would be \[\frac{ 1-3 }{ 0-4 } = \frac{ -2 }{ -4}=\frac{ 1 }{ 2 }\]
Like I said earlier, use the slope of the segment to find the opposite reciprocal. Which would be \[\frac{ 1 }{ 2 } \rightarrow -\frac{ 2 }{ 1 }\] this is the slope of the perpendicular line. But because it bisects the segment, we must also find the midpoint.

\[(\frac{ x1+x2 }{ 2 }, \frac{ y2+y1 }{ 2 })\]

So

\[\frac{ 1+3 }{ 2 }=2\]
and \[\frac{ 0+4 }{ 2 }=2\]

So the midpoint is (2,2)

Now we use point slope form to find the actual equation of the perpendicular bisector.

\[y-y1=m(x-x1)\] (recall, y1 and x1 are from the midpoint.)

\[y-2=-2(x-2)\]
Distribute
\[y-2=-2x+4\]
add 2 to both sides to get y alone.
\[y=-2x+4\]

Just to reverify that it is, find the equation of the original line, also using point slope form.

y-0=\[y-0=\frac{ 1 }{ 2 }(x-1)\]

\[y=\frac{ 1 }{ 2 }x- \frac{ 1 }{ 2 }\]