Question

A triangle has corners A, B, and C located at `(4 ,5 )`, `(7 ,1 )`, and `(2 ,3 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

End point of altitude through `C` is at `(4.24,4.68)`
Length of the altitude is `2.8`

Explanation:

Given
`color(white)("XXX")A" at " (4,5)`
`color(white)("XXX")B" at " (7,1)` and
`color(white)("XXX")C" at " (2,3)`

`"Slope"_(AB) = (5-1)/(4-7) =-4/3`
and the equation of the line through `A` and `B` can be written as
`color(white)("XXX")(y-5)=-4/3(x-4)`
or
`color(white)("XXX")4x+3y=31color(white)("XXXX")[1]`

The altitude going through `C` has a base of `AB`
and is perpendicular to `AB`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`rarr`The equation of the altitude line through `C` has a slope of `3/4`
and since it passes through `C=(2,3)`
its equation can be written as
`color(white)("XXX")y-3=3/4(x-2)`
or
`color(white)("XXX")3x-4y=-6color(white)("XXXX")[2]`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

[1]`color(white)("XXX")4x+3y=31color(white)("XXXX")`(the base line: AB)
[2]`color(white)("XXX")3x-4y=-6color(white)("XXXX")`(the altitude line)

Multiplying [1] by `4`
and [2] by `3`
[3]`color(white)("XXX")16x+12y=124`
[4]`color(white)("XXX")9x-12y=-18`

Adding [3] and [4]
[5]`color(white)("XXX")25x= 106`

[6]`color(white)("XXX")x=106/25=4.24`

Substituting `106/25` for `x` back in [1]
[7]`color(white)("XXX")4(106/25)+3y=31`

[8]`color(white)("XXX")424/25+3y=775/25`

[9]`color(white)("XXX")3y=351/25`

[10]`color(white)("XXX")y=117/25=4.68`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Length of the altitude
`color(white)("XXX")=sqrt((4.24-2)^2+(4.68-3)^2`

`color(white)("XXX")=2.8`