HELP!! Geometry question!?

Question

Find the coordinates of the ortho center of triangle abc: A (0,0) B(5,0) C(5,3) 

Answer 1

The points are A (0,0)  B(5,0)  C(5,3) 

First find the slopes of AB,BC,CA

Slope of AB= (0 - 0) /( 5 - 0) = 0.

Slope of BC = (3 - 0) / (5 - 5) =3/ 0 

Slope of CA = (3 - 0) / (5 - 0) = 3 / 5 .

Now let's find the altitudes of AD,BE,CF which are perpendicular to BC, CA,AB

Since Product of the slopes = -1

Slope of AD = -1 / BC = -1 /     = 0

Slope of BE = -1 / CA = - 1 / (3/5) = -5 / 3 .

Slope of CF =   - 1 / AB =  - 1  / 0 .

Now the equation of the lines for AD ,BE , CF

Equation of line for AD:

A(0 , 0)  with slope 0

 y - y‚ = m(x - x‚) =y - 0 = 0=>y  = 0.  - - - - >(1)

Equation of line for BE :

B( 5, 0 ) and m = - 5 / 3 .

 y - y‚ = m(x - x‚) =y - 0 =  - 5 / 3 (x - 5)=3=> y = -5 x + 25

5x + 3y - 25 = 0 - - - - >(2)

Equation of the line for CF :

C(5 , 3) and  m =  -1 / 0

y - 3 = -1/0(x - 5)

x = 5.  - - - >(3)

Hence (x , y ) = (5, 0)

Hence the co -ordinates of the orthocentre are (x , y ) = (5, 0) .