A triangle has corners A, B, and C located at #(4 ,8 )#, #(7 ,4 )#, and #(5 ,3 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer
Jun 24, 2016

End points of the altitude are #(5,3) & (6.76,4.32)#
Length of altitude #2.2# unit**

Explanation:

Let CD be the altitude drawn from C perpendicular to side AB touches at D of side AB.
Slope of line AB is #(4-8)/(7-4) = -4/3 :.#slope of line CD is # -1/(-4/3) =3/4# Equation of line CD is #y-3=3/4(x-5) or 4y-12=3x-15 or 4y-3x= -3; (1)# Equation of line AB is #y-8 = -4/3(x-4) or 3y-24 =-4x +16 or 3y+4x = 40; (2) #
Now solving equation (1) & equation (2) we find the co-ordinate of poin D. #12y-9x = -9 # (1)x3 (3)
#12y+16x = 160# (2)x4 (4)
subtracting (3) from (4) we get #25x=169 or x =169/25=6.76 :. y =(40-(4*6.76))/3 =4.32# End points of the altitude are #(5,3) & (6.76,4.32)#
Length of altitude #= sqrt((5-6.76)^2+(3-4.32)^2) =2.2#[Ans]