You don't need the Pythagorean theorem.

Given ߡABC with vertices A(-1,2), B(5,-4), C(9,6), prove that CD
is both an altitude (perpendicular to base AB) and a median
(bisects the base) where D is the point D(-4,3).

Draw ߡABC and AD



We need to show that 

(1) CD ⊥ AB  and  (2) D is the midpoint of AB.

For (1), we find the slopes of CD and AB and show that they are negative
reciprocals, that their product is -1.

Slope formula:

m = 

To find the slope of CD:
where (x1,y1) = C(6,8)
and where (x2,y2) = D(-4,3)

m =  =  = 

To find the slope of AB:
where (x1,y1) = A(-6,7)
and where (x2,y2) = B(-2,-1)

m =  =  =  = -2

Since  and  are negative reciprocals,
their product is -1, we have proved that CD ⊥ AB,
which proves that CD is an altitude to base AB.

For (2), we use the midpoint formula to show that D(-4,3) is the
midpoint of AB.

Midpoint formula:

Midpoint = 

where (x1,y1) = A(-6,7)
and where (x2,y2) = B(-2,-1)

Midpoint =  =  = (-4,3) which is point D.

Thus CD is also a median to the base AB of ߡABC.

Edwin