A triangle has corners A, B, and C located at #(8 ,3 )#, #(4 ,5 )#, and #(2 , 7 )#, 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   ,3   )#, #(4  ,5    )#, and #(2 ,  7   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Avery Allen · Accepted Answer

End points of the altitude are #(8/5,31/5), (2,7)#
Length of altitude #= 2sqrt(1/5)=0.894#

As shown in the diagram, #CD# is the altitude perpendicular to #AB# from point #C#.

The slope of #AB= (5-3)/(4-8)=-1/2#
#=># The slope of #CD=2# (perpendicular to #AB)#

The equation of #AB# is : #y-3=(-1/2)(x-8)#
#=> y+x/2=7 ..........(1)#
The equation of #CD# is : #y-7=2(x-2)#
#=> y-2x=3 ............ (2)#

Solving (1) and (2) we get :
#x=8/5, y=31/5#

So endpoints of altitude #CD=(8/5,31/5) and (2,7)#

Length of altitude #CD= sqrt((2-8/5)^2+(7-31/5)^2)#
#=sqrt(4/5)=2sqrt(1/5)=0.894#

Mason Adams · Answer

undefined The endpoints of the altitude going through corner C are the coordinates of C and the foot of the altitude, which we'll denote as D. To find the foot of the altitude, we can use the formula for the altitude of a triangle from a given vertex:

$ x_D = \frac{{x_A + x_B + x_C}}{3} $ and $ y_D = \frac{{y_A + y_B + y_C}}{3} $

Once we have the coordinates of D, we can calculate the length of CD using the distance formula:

$ CD = \sqrt{{(x_C - x_D)^2 + (y_C - y_D)^2}} $