What is the orthocenter of a triangle with corners at #(1 ,3 )#, #(6 ,9 )#, and (2 ,4 )#?

1 Answer
Aug 5, 2018

The orthocenter of triangle is #(56,-41)#

Explanation:

Let #triangleABC " be the triangle with corners at"#

#A(1,3), B(6,9) and C(2,4)#

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) =(3-9)/(1-6)=6/5#

#bar(AB)_|_bar(CN)=>#slope of # bar(CN)=-5/6# ,

# bar(CN)# passes through #C(2,4)#

#:.#The equn. of #bar(CN)# is #:y-4=-5/6(x-2)#

#6y-24=-5x+10#

#i.e. color(red)(5x+6y=34.....to (1)#

Slope of #bar(BC) =(9-4)/(6-2)=5/4#

#bar(AL)_|_bar(BC)=>#slope of # bar(AL)=-4/5# , # bar(AL)# passes through #A(1,3)#

#:.#The equn. of #bar(AL)# is #:y-3=-4/5(x-1)#

#=>5y-15=-4x+4#

#=>color(red)(4x+5y=19.....to (2)#

Taking #eqn.(1)xx5-eqn.(2)xx(-6)# and adding

#color(white)(....)25x+30y=170#
#ul(-24x-30y=-114#
#:.1x-0=56#

#=>color(blue)( x=56#

From equn.#(2)# we get

#4(56)+5y=19=>5y=19-224=-205=>color(blue)(y=-41#

Hence, the orthocenter of triangle is #(56,-41)#