Question

A triangle has corners A, B, and C located at `(5 ,6 )`, `(3 ,9 )`, and `(1 , 2 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

The end points are `=(73/13,198/39)` and the length of the altitude is `=5.55`

Explanation:

The corners of the triangle are

`A=(5,6)`

`B=(3,9)`

`C=(1,2)`

The slope of the line `AB` is `m=(9-6)/(3-5)=-3/2`

The equation of line `AB` is

`y-9=-3/2(x-3)`

`y-9=-3/2x+9/2`

`y+3/2x=27/2`...........................`(1)`

`mm'=-1`

The slope of the line perpendicular to `AB` is `m'=2/3`

The equation of the altitude through `C` is

`y-2=2/3(x-1)`

`y-2=2/3x-2/3`

`y=2/3x+4/3`................................`(2)`

Solving for `x` and `y` in equations `(1)` and `(2)`, we get

`-3/2x+27/2=2/3x+4/3`

`2/3x+3/2x=27/2-4/3`

`13/6x=73/6`

`x=73/13`

`y=2/3*73/13+4/3=198/39`

The end points of the altitude is `=(73/13,198/39)`

The length of the altitude is

`=sqrt((1-73/13)^2+(2-198/39)^2)`

`=sqrt((-60/13)^2+(-120/39)^2)`

`=sqrt(46800)/39`

`=5.55`