Question

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

Answer 1

Endpoints of altitude are `(59/17,104/17), (1,2)`
Length of altitude `=sqrt(392/17)~~4.802`

Explanation:

As shown in the diagram, line `CD` is the altitude perpendicular to line `AB` from point `C`

Given `A(2,7), B(7,4), C(1,2)`,
slope of line `AB = (4-7)/(7-2)=-3/5`
equation of line `AB` is : `y-4=-3/5(x-7)`
`=> 5y=-3x+41` ..... (1)

Let slope of line `AB` be `s_1`, and slope of line `CD` be `s_2`
As line `CD` is perpendicular to line `AB`, `=> s_1*s_2=-1`
`=> -3/5*s_2=-1, => s_2=5/3`
So slope of line `CD= 5/3`
equation of line `CD` is : `(y-2)=5/3(x-1)`
`=> 3y=5x+1` ..... (2)

Solving (1) and (2) we get `x=59/17, y=104/17`
Hence, endpoints of the altitude are `(59/17, 104/17) and (1,2)`

length of altitude `=sqrt((59/17-1)^2+(104/17-2)^2`

`= sqrt(392/17)~~4.802`