What is the orthocenter of a triangle with corners at #(9 ,7 )#, #(2 ,4 )#, and (8 ,6 )#?

Question

What is the orthocenter of a triangle with corners at #(9  ,7   )#, #(2   ,4   )#, and (8    ,6   )#?

Aurora Ayala · Accepted Answer

The orthocenter of triangle is #(14,-8)#

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

#A(9,7), B(2,4) and C(8,6)#

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 .

Slope of #bar(AB) =(7-4)/(9-2)=3/7#

#bar(AB)_|_bar(CN)=>#slope of # bar(CN)=-7/3# ,

# bar(CN)# passes through #C(8,6)#

#:.#The equn. of #bar(CN)# is #:y-6=-7/3(x-8)#

#3y-18=-7x+56#

#i.e. color(red)(7x+3y=74.....to (1)#

Slope of #bar(BC) =(6-4)/(8-2)=2/6=1/3#

#bar(AL)_|_bar(BC)=>#slope of # bar(AL)=-3# , # bar(AL)# passes through #A(9,7)#

#:.#The equn. of #bar(AL)# is #:y-7=-3(x-9)=>y-7=-3x+27#

#=>3x+y=34#

#i.e. color(red)(y=34-3x.....to (2)#

Subst. #color(red)(y=34-3x# into #(1)# ,we get

#7x+3(34-3x)=74=>7x+102-9x#=#74=>-2x=-28#
#=>color(blue)( x=14#

From equn.#(2)# we get

#y=34-3(14)=34-42=>color(blue)(y=-8#

Hence, the orthocenter of triangle is #(14,-8)#

Sadie Allen · Answer

undefined To find the orthocenter of a triangle, you need to find the point where the altitudes of the triangle intersect. The altitude of a triangle is a perpendicular line segment from a vertex to the line containing the opposite side. To find the orthocenter, you can first find the equations of the altitudes and then solve for their point of intersection.

Given the coordinates of the triangle's vertices as (9, 7), (2, 4), and (8, 6), you can find the equations of the altitudes and then solve for their point of intersection, which will be the orthocenter.