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

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Answer 1 · 2016-10-24T11:26:01

Given in triangle ABC the coordinates of A,B,C are

#A->(6,8),B-> (7,4),C-> (5,3)#

Let D (h,k) be the foot of the perpedicular drawn from C (5,3) on AB of the given triangle ABC.

So CD is the altitude going through corner C.

#m_"AB"->"The slope of AB"=(8-4)/(6-7)=-4#

#m_"AD"->"The slope of AD"=(k-8)/(h-6)#

#m_"CD"->"The slope of CD"=(k-3)/(h-5)#

Now AB and AD are on the same straight line
So
#m_"AD"=m_"AB"#

#=>(k-8)/(h-6)=-4#

#=>k=-4h+24+8=-4h+32....(1)#

Again CD is perpendicular on AB

So
#m_"CD"xx m_"AB"=-1#

#=>(k-3)/(h-5)xx(-4)=-1#

#=>k=(h-5)/4+3=(h+7)/4.....(2)#

comparing (1) and (2)

#(h+7)/4=-4h+32#

#=>h+7=-16h+128#

#=>17h=121#

#=>h=121/17#

plugging in the value of h in (1)

#k=-4xx121/17+32=(544-484)/17=60/17#

So the coordinates of the end points of altitude CD

#C->(5,3) and D->(121/17,60/17)#