What is the orthocenter of a triangle with corners at #(6 ,3 )#, #(4 ,5 )#, and (2 ,9 )#?

1 Answer
Jul 2, 2018

Answer:

The orthocenter of triangle is #(-14,-7)#

Explanation:

Let #triangle ABC# be the triangle with corners at

#A(6,3) ,B(4,5) and C(2,9)#

Let #bar(AL) , bar(BM) and bar(CN) # be the altitudes of sides

#bar(BC) ,bar(AC) ,and bar(AB)# respectively.

Let #(x,y)# be the intersection of three altitudes .

enter image source here

Slope of #bar(AB) =(5-3)/(4-6)=-1#

#bar(AB)_|_bar(CN)=>#slope of # bar(CN)=1# , # bar(CN)# passes through #C(2,9)#

#:.#The equn. of #bar(CN)# is #:y-9=1(x-2)#

#i.e. color(red)(x-y=-7.....to (1)#

Slope of #bar(BC) =(9-5)/(2-4)=-2#

#bar(AL)_|_bar(BC)=>#slope of # bar(AL)=1/2# , # bar(AL)# passes through #A(6,3)#

#:.#The equn. of #bar(AL)# is #:y-3=1/2(x-6)=>2y-6=x-6#

#i.e. color(red)(x=2y.....to (2)#

Subst. #x=2y# into #(1)# ,we get

#2y-y=-7=>color(blue)( y=-7#

From equn.#(2)# we get

#x=2y=2(-7)=>color(blue)(x=-14#

Hence, the orthocenter of triangle is #(-14,-7)#