SOLUTION: how do i prove an altitude? What are the steps to doing this in a coordinate geometry question where they ask you to prove that line CD in triangle ABC is an altitude and is bisect

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Question 721940: how do i prove an altitude? What are the steps to doing this in a coordinate geometry question where they ask you to prove that line CD in triangle ABC is an altitude and is bisecting the base of the triangle.
Found 2 solutions by mananth, Edwin McCravy:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
CD is bisecting the base
find co-ordinates of D by mid point formula.
You have to prove CD as altitude.
Apply Pythagoras theorem.
show that AC^2-AD^2=CD^2
by distance formula

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
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 = %28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29

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

m = %28%283%29-%288%29%29%2F%28%28-4%29-%286%29%29 = -5%2F-10 = 1%2F2

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

m = %28%28-1%29-%287%29%29%2F%28%28-2%29-%28-6%29%29 = -8%2F%28-2%2B6%29 = %28-8%29%2F4 = -2

Since 1%2F2 and -2 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 =  = %28matrix%281%2C3%2C%28-8%29%2F2%2C+%22%2C%22%2C%286%29%2F2%29%29 = (-4,3) which is point D.

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

Edwin