A triangle has corners A, B, and C located at #(7 ,6 )#, #(4 ,3 )#, and #(5 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer
Jim G.
Jul 30, 2018

#6,7),(5,8),2sqrt2#

Explanation:

#"the altitude from C meets AB the side opposite at right"#
#"angles"#

#"let D be the point on AB where the altitude intersects AB"#

#m_(AB)=(6-3)/(7-4)=3/3=1#

#"hence "m_(CD)=-1/m_(AB)=-1#

#color(blue)"equation of AB"#

#"using "m=1" and "(a,b)=(4,3)" then"#

#y-3=x-4rArry=x-1to(1)#

#color(blue)"equation of CD"#

#"using "m=-1" and "(a,b)=(5,8)" then"#

#y-8=-(x-5)#

#y=-x+13to(2)#

#"solving "(1)" and "(2)#

#x-1=-x+13rArrx=7#

#y=7-1=6#

#"coordinates of D "=(7,6)#

#"thus the endpoints of the altitude are "(7,6)" and "(5,8)#

#d=sqrt((5-7)^2+(8-6)^2)=sqrt(4+4)=sqrt8=2sqrt2#

#"length of altitude "=2sqrt2#

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