Question

What is the orthocenter of a triangle with corners at `(9 ,5 )`, `(4 ,4 )`, and (8 ,6 )#?

Answer 1

Ortho center (23/3, 23/3)

Explanation:

Slope of AB `=(4-5)/(4-9) = 1/5`
Slope of CO perpendicular to AB is -5 (O is the ortho center)
Eqn of OC is
`y-6=-5(x-8)`
`5x+y = 46color(white)(aaa) Eqn (1)`

Slope of AC`=(6-5)/8-9) = -1`
Slope of OB perpendicular to AC is `-(1/-1)=1`
Eqn of OB is
`y-4= 1*(x- 4)`
`-x+y=0 color(white)(aaa)`Eqn (2)

Solving Eqns (1) & (2),
`6x = 46, x= 23/3`
`y= 23/3`
Coordinates of O (8,6)#

Verification :
Slope of BC `= (6-4)/(8-4) = 1/2`
Slope of OA `=-2`
Eqn of OA is
`y - 5 = -2(x - 9)`
`2x+y = 23 color (white)(aaa)` Eqn (3)

Solving Eqn (2) & (3),
`3x = 23, x=23/3, y = 23/3`