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

Question

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

Mason Adams · Accepted Answer

Hence, the orthocenter of triangle is #(157/7,-23/7)#

Let #triangle ABC# be the triangle with corners at

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

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) =(9-4)/(4-3)=5#

#bar(AB)_|_bar(CN)=>#slope of # bar(CN)#=#-1/5# , # bar(CN)# passes through #C(1,1)#

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

#=>5y-5=-x+1#

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

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

#bar(AL)_|_bar(BC)=>#slope of # bar(AL)=-2/3# , # bar(AL)# passes through #A(4,9)#

#:.#The equn. of #bar(AL)# is #:y-9=-2/3(x-4)=>3y-27=-2x+8#

#i.e. color(red)(2x+3y=35.....to (2)#

Subst. #x=6-5y# into #(2)# ,we get

#2(6-5y)+3y=35#

#=>-7y=23#

#=>color(blue)( y=-23/7#

From equn.#(1)# we get

#x=6-5(-23/7)=(42+115)/7=>color(blue)(x=157/7#

Hence, the orthocenter of triangle is #(157/7,-23/7)#

Allison Allen · Answer

undefined To find the orthocenter of a triangle, follow these steps:

1. Calculate the slopes of the lines passing through each pair of vertices to determine the slopes of the altitudes.
2. Use the point-slope form of a line to find the equations of the altitudes.
3. Solve the system of equations formed by the altitude equations to find the intersection point, which is the orthocenter.