A triangle has corners A, B, and C located at #(5 ,6 )#, #(3 ,5 )#, and #(1 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-07-01T04:33:24

The endpoint is #(1/5,18/5)#
The length of altitude is #=1.79#

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The altitude going through #C(1,2)# is perpendicular to line #AB#, as shown in the figure.

Let the endpoint of the altitude be #D#.

The slope of line #AB=(5-6)/(3-5)=1/2#
Hence, the slope of the required altitude #=-2#

The equation of the altitude (line #CD#) is :
#y-2=-2(x-1)#
#y-2=-2x+2#
#2x+y=4# ---------------(1)

The equation of line #AB# is :
#y-5=1/2(x-3)#
#2y-10=x-3#
#2y-x=7# ----------------(2)

Solving for #(x,y)# in EQ (1) and (2), we get the end point of the altitude #D#,
The endpoint is #(x,y)=(1/5, 18/5)=(0.2,3.6)#

Length of altitude = length of #CD#
#=sqrt((0.2-1)^2+(3.6-2)^2)=sqrt(3.2)=1.79#