Question

A triangle has corners A, B, and C located at `(3 ,4 )`, `(7 ,5 )`, and `(2 ,9 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

the endpoints and length of the altitude going through corner C happen to be `(x,y)=(0.529,3.118)`

Explanation:

Equation of the line`AB` joining `A(3,4), and B(7,5)` is given by
`(y-4)/(x-3)=(5-y)/(7-x)`
Simplifying
`(y-4)(7-x)=(5-y)(x-3)`
`7y-28-xy+4x=5x-15-xy+3y`
`4x-5x+3y-7y-15+28=0`
`-x-4y+13=0`
Expressing in the standard form
`ax+by+c=0`,
`x+4y-13=0`
Comparing,
`a=1, b=4, c==13`
Slope of the side `AB` is
`m=-a/b=-1/4=-1/4`
Slope of the perpendicular to side `AB` is
`m'=-1/m=-1/(-1/4)=4`
Equation of the perpendicular through `C(2,9)` is given by
`(y-9)/(x-2)=4`
Simplifying
`y-9=(x-2)(4)`
`y-9=4x-8`
`4x-y-8+9=0=0`
`4x-y+1=0`
The intersection of the two lines
`x+4y-13=0` and
`4x-y+1=0`

Solving for `x and y`

`x+4y=13`
`4x-y=-1`

`x=13-4y`
`4(13-4y)-y=-1`
`52-16y-y=-1`
`17y=53`
`y=53/17=3.118`
`x=13-4(3.118)`
`x=0.529`

the endpoints and length of the altitude going through corner C happen to be `(x,y)=(0.529,3.118)`
Check
`0.529+4(3.118)-13=0.001`
`4(0.529)-3.118+1=-0.002`