A triangle has corners A, B, and C located at #(2 ,2 )#, #(3 ,4 )#, and #(6 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-01-30T19:48:16

#"The endpoints of altitude through corner C are:"C(6,8) " and "D(5.2,8.4)#

#"The length of altitude is : "0.89#

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#"The slope of line through A(2,2) and B(3,4) is :" #

#m=(4-2)/(3-1)=2/1=2#

#"now,let us find the slope of line passing C(6,8) and D(x,y)"#

#"if two line is perpendicular , product of their slopes is equal to -1"#

#m*n=-1#
#2*n=-1#
#n=-1/2#

#"now ,let us write equations of lines "#

#"equation for line passing A(2,2) and D(x,y)"#
#y-2=2(x-2)#
#y=2x-4+2#
#y=2x-2" (1)"#

#"equation for line passing C(6,8) and D(x,y)"#
#y-8=-1/2(x-6)#
#y=-1/2 (x-6)+8#

#y=-1/2 x+3+8#

#y=-1/2 x+11" (2)"#

#"let us write the equations (1) and (2) as equal "#

#2x-2=-1/2 x+11#

#2x+1/2 x=11+2#

#5/2 x=13" , "5x=13*2" ," x=26/5" , "x=5.2#

#"use equation (1) or (2)"#

#y=2x-2#

#"insert x=5.2"#

#y=2*5.2-2#
#y=10.4-2#
#y=8.4#

#D=(5.2,8.4)#

#"the end points are C(6,8) and D(5.2,8.4)"#

#"The length of the altitude can be calculated using :"#

#h_C=sqrt((5.2-6)2+(8.4-8)^2)#

#h_C=sqrt((-0.8)+(0.4)^2)#

#h_C=sqrt(0.64+0.16)#

#h_C=sqrt(0.80)#

#h_C=0.89#