Question

A triangle has corners A, B, and C located at `(1 ,6 )`, `(9 ,3 )`, and `(2 ,2 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

The altitude endpoints are `(2,2)` and `(3.1918,5.1781)`
The altitude has a length of `3.394`

Explanation:

The line segment `AB` through `A=(1,6)` and `B=(9,3)`
has a slope of `m_(AB)=(6-3)/(1-9)=-3/8`
and an equation of `y-6=-3/8(x-1)`
or
`color(white)("XXX")y=(-3x+51)/8`

In `triangleABC` the altitude through `C` is a line segment perpendicular to `AB` and therefore has a slope of
`color(white)("XXX")m_"alt"=8/3`

Since the altitude passes through `C=(2,2)` and has a slope of `m_"alt"=8/3`
it's line has an equation:
`color(white)("XXX")y-2=8/3(x-2)`
or
`color(white)("XXX")y=(8x-10)/3`

Using standard methods and a calculator
`color(white)("XXX")AB: y=(-3x+51)/8`
and
`color(white)("XXX")"Altitude": y=(8x-10)/3`
intersect at `(3.1918,5.1781)` (approximate)

The length of the altitude from `C: (2,2)` to the point of intersection is
`color(white)("XXX")d=sqrt((2-3.1918)^2+(2-5.1781)^2)~~5.1781`