SOLUTION: Given: triangle ABC A(-1,2) B(7,0) C(1,-6) and a point D(4,-3) on segment BC Prove: segment AD is the perpendicular bisector of segment BC

 

First we use the midpoint formula to show that D 
is the midpoint of BC.  That will show that AD is 
a bisector of BC.  Then we will use the slope 
formula to show that AD is perpendicular to BC.

The midpoint of the segment joining (,) and (,) is
given by the formula


 = (,)

We use B(,) as (,)
and C(,) as (,) 

 = (,)

 = (,)

 = (,)  

 = (,)

Since D has those coordinates, AD bisects BC.

Now we need to show AD and BC are perpendicular.

The slope of the segment joining (,) and (,) is
given by the formula



So we now find the slope of BC, again using
B(,) as (,)
and C(,) as (,)









So the slope of BC is 

Now So we now find the slope of AD, using
A(,) as (,)
and D(,) as (,)









So the slope of AD is 

Since  and  are reciprocals with 
opposite signs, this proves AD is perpendicular to
BC.

Therefore AD is the perpendicular bisector of BC.

Edwin