How do you find the equation of a triangles altitude using point slope formula?

1. An altitude of a triangle is a line segment from a vertex perpendicular to the opposite side. Find the equations of the altitudes of the triangle with vertices (4, 5),(-4, 1) and (2, -5). Do this by solving a system of two of two of the altitude equations and showing that the intersection point also belongs to the third line. 
(Scroll Down for Answer!)Answer by jim_thompson5910(34047)   (Show Source):You can put this solution on YOUR website!
If we plot the points and connect them, we get this triangle: 

 

Let point 
A=(xA,yA)
B=(xB,yB)
C=(xC,yC) 

 
Let's find the equation of the segment AB 

Start with the general formula 

 

Plug in the given points 

 

Simplify and combine like terms 

 

So the equation of the line through AB is  

 
Let's find the equation of the segment BC 

Start with the general formula 

 

Plug in the given points 

 

Simplify and combine like terms 

 

So the equation of the line through BC is  

 
Let's find the equation of the segment CA 

Start with the general formula 

 

Plug in the given points 

 

Simplify and combine like terms 

 

So the equation of the line through CA is  

So we have these equations of the lines that make up the triangle 

 

So to find the equation of the line that is perpendicular to  that goes through the point C(2,-5), simply negate and invert the slope  to get 

Now plug the slope and the point (2,-5) into  

 

 Solve for y and simplify 

So the altitude for vertex C is  

Now to find the equation of the line that is perpendicular to  that goes through the point A(4,5), simply negate and invert the slope to get  

Now plug the slope and the point (2,-5) into  

 

 Solve for y and simplify 

So the altitude for vertex A is  

Now to find the equation of the line that is perpendicular to  that goes through the point B(-4,1), simply negate and invert the slope to get  

Now plug the slope and the point (-4,1) into  

 

 Solve for y and simplify 

So the altitude for vertex B is  

 
Now let's solve the system 

 

 Plug in  into the first equation 

 Add 2x to both sides and subtract 2 from both sides 

 Divide both sides by 3 to isolate x 

Now plug this into  

 

 

So the orthocenter is (-2/3,1/3) 

So if we plug in  into the third equation , we get 

 

 

 

 

So the orthocenter lies on the third altitude 

There was 504 pages in that chunk