A triangle has corners A, B, and C located at #(8 ,7 )#, #(4 ,5 )#, and #(6 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

Question

A triangle has corners A, B, and C located at #(8   ,7   )#, #(4  ,5    )#, and #(6 ,  2   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Allison Allen · Accepted Answer

The endpoint is #=(4.4,5.2)#
The length of the altitude is #=3.58# The triangle's corners are #A=(8,7)# #B=(4,5)# #C=(6,2)# The slope of #AB# is #=(7-5)/(8-4)=2/4=1/2# The slope of the line through #C# and perpendicular to #BC# is #=-2# The altitude's equation is #(y-2)=-2(x-6)# #y-2=-2x+12# #2x+y=14#....................#(1)# The equation of the line #AB# is #y-5=1/2(x-4)# #2y-10=x-4# #2y-x=6#......................#(2)# Solving for #(x,y)# in equations #(1)# and #(2)#, we get the end point of the altitude #2x+((x+6))/2=14# #4x+6+x=28# #5x=22# #x=22/5=4.4# #y=14-2*22/5==5.2# The endpoint is #=(4.4,5.2)# The altitude's duration is #=sqrt((6-4.4)^2+(2-5.2)^2)# #=sqrt(1.6^2+(-3.2)^2)# #=sqrt(12.8)# #=3.58#

Ella Armstrong · Answer

undefined The endpoints of the altitude going through corner C are (6,2) and the foot of the altitude, which can be found by calculating the intersection point of the line containing AB and the perpendicular bisector of segment AB passing through C. The length of the altitude can be calculated using the distance formula between the endpoints.