#y-1=4(x-2)# equation of the line containing A and B
#y-8=-1/4(x-4)# equation of the line containing C and perpendicular to line AB
Solve for the other endpoint of the altitude
#y=4x-7" "# equation of the line containing A and B after simplification
#x+4y=36" "#equation of the line containing C and perpendicular to line AB after simplification
Simultaneous solution
#x+4(4x-7)=36" "#
#17x=36+28#
#17x=64#
#x=64/17#
#y=4x-7#
#y=4(64/17)-7#
#y=(256-119)/17#
#y=137/17#
The other endpoint is #(64/17, 137/17)#
the length #l#
#l=sqrt((64/17-x_c)^2+(137/17-y_c)^2)#
#l=sqrt((64/17-4)^2+(137/17-8)^2)#
#l=sqrt(((64-68)/17)^2+((137-136)/17)^2)#
#l=sqrt(((-4)/17)^2+((1)/17)^2)#
#l=sqrt(((-4)/17)^2+((1)/17)^2)#
#l=1/17sqrt(17)=0.2425#
God bless....I hope the explanation is useful.
