Question

What is the orthocenter of a triangle with corners at `(9 ,7 )`, `(6 ,9 )`, and (2 ,4 )#?

Answer 1

Orthocenter is at `(132/23,221/23)~~(5.74,9.61)`
`color(white)("XXX")`...based on the outrageous assumption that my arithmetic is valid

Explanation:

For convenience label the points on this triangle as in the diagram below:

Note that (by definition) the orthocenter of a triangle is the point where the altitudes meet (labelled `O` in this diagram).

Slope(`AB`)`=(7-4)/(9-2)=3/7`
Slope(`CD`)`=-7/3` (since it is perpendicular to `AB`)
Equation of line (`CD`) [using point slope form is]
`color(white)("XXX")y-9=-7/3(x-6)`
`color(white)("XXX")3y-27=-7x+42`
`color(white)("XXX")7x+3y=69`

Slope(`CB`)`=(9-7)/(6-9)=-2/3`
Slope(`AE`)`=3/2`
Equation of line (`AE`)
`color(white)("XXX")y-4=3/2(x-2)`
`color(white)("XXX")2y-8=3x-6`
`color(white)("XXX")3x-2y=-2`

Solving the system of equations:
`{([1],,7x+3y=69),([2],,3x-2y=-2):}`

`rArr {([1]xx2,,14x+6y=138),([2]xx3,,9x-6y=-6):}`

`rarr 23x=132`
`rarr x=132/23`

and
`rArr {([1]xx3,,21x+9y=207),([2]xx7,,21x-14=-14):}`

`rarr 23y=221`
`rarr y=221/23`