Question

Triangle abc has vertices a(0 0) b(6 8) and c(8 4). which equation represents the perpendicular bisector of bc

Answer 1

Perpendicular means that the slope of one line is the opposite reciprocal of the other; bisector means that the line that is perpendicular is cutting the other one exactly in half.  So we have to find 2 things about BC:  the first is the midpoint which will then serve as the x and y coordinates in our new equation, and second is the slope of it.  First things first:  the midpoint formula is this: $M=( \frac{ x_{1}+ x_{2} }{2} , \frac{ y_{1} + y_{2} }{2} )$.  Using our coordinates we fill in that formula like this:  $M=( \frac{6+8}{2}, \frac{8+4}{2} )$ which of course gives us a midpoint of (7, 6).  Now for the slope of the line BC:  $m= \frac{4-8}{8-6} =-2$.  But since we need the perpendicular slope of that we will take its opposite reciprocal which is 1/2.  Now use the x and y coordinates of the midpoint and that newly found slope and write your equation to solve for b. $6= \frac{1}{2}(7)+b$.  Solving that for b gives you that b = 5/2.  So the equation of the line that is the perpendicular bisector of BC (drum roll, please...) is  $y= \frac{1}{2} x+ \frac{5}{2}$