A triangle has corners A, B, and C located at #(2 ,7 )#, #(3 ,5 )#, and #(9 , 6 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer
CW
Dec 28, 2016

endpoints of altitude are #(19/5,17/5), and (9,6)#
length of altitude #=sqrt(169/5)=5.8138#

Explanation:

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

Given #A(2,7), B(3,5), C(9,6)#,
slope of line #AB = (5-7)/(3-2)=-2#
equation of line #AB= (y-5)=-2(x-3)#
#=> y=-2x+11 ..... (1)#
Let slope of line #AB# be #s_1#, and slope of line #CD# be #s_2#
As line #CD# is perpendicular to line #AB#, #=> s_1*s_2=-1#
#=> -2*s_2=-1, => s_2=1/2#
slope of line #CD= 1/2#
equation of line #CD=(y-6)=1/2(x-9)#
#=> y=1/2x+3/2 .....(2)#

solving (1) and (2) we get #x=19/5, y=17/5#
So endpoints of the altitude are #(19/5,17/5) and (9,6)#

length of altitude #=sqrt((9-19/5)^2+(6-17/5)^2#

#= sqrt(169/5)=5.8138#

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