A triangle has corners A, B, and C located at #(6 ,1 )#, #(2 ,4 )#, and #(3 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer
Sep 30, 2017

End points of altitude CF are #(3,2) and (54/15) , (14/5)#
Length of altitude CF = 1

Explanation:

Equation of side AB is
#(y-y1)/(y2-y1)=(x-x1)/(x2-x1)#
#(y-1)/(4-1)=(x-6)/(2-6)#
#3x+4y=22# (Equation 1)
Slope of AB m1#=(y2-y1)/(x2-x1)=(4-1)/(2-6)=-(3/4)#
Slope of altitude CF
#m2=-1/(m1)=-1/(-3/4)=4/3#
Equation of altitude CF
#(y-y3)=m2(x-x3)#
#(y-2)=(4/3)(x-3)#
#4x-3y=6# (Equation 2)
By solving Equation 1 & 2, we get point F
#12x+16y=88#
#12x-9y=18#
Subtracting, #25y=70#
#y=(14/5)#
Substituting value of y in Equation 1, we get #x=54/15#
Coordinates of #F (54/15,14/5)#
Length of altitude CF
#CF=sqrt(((14/5)-2)^2+((54/15)-3)^2)#
#CF=sqrt((4/5)^2+(9/15)^2)=sqrt((144+81)/225)=1#