What is the orthocenter of a triangle with corners at #(6 ,3 )#, #(2 ,4 )#, and (7 ,9 )#?

1 Answer
Jan 20, 2018

Answer:

Orthocenter of the triangle is at #(5.6,3.4) #

Explanation:

Orthocenter is the point where the three "altitudes" of a triangle meet. An "altitude" is a line that goes through a vertex (corner point) and is at right angles to the opposite side.

#A = (6,3) , B(2,4) , C(7,9) # . Let #AD# be the altitude from #A# on #BC# and #CF# be the altitude from #C# on #AB# they meet at point #O# , the orthocenter.

Slope of #BC# is #m_1= (9-4)/(7-2)=5/5= 1#

Slope of perpendicular #AD# is #m_2= -1 (m_1*m_2=-1) #

Equation of line #AD# passing through #A(6,3)# is

#y-3= -1(x-6)or y-3 = -x+6 or x +y = 9 (1)#

Slope of #AB# is #m_1= (4-3)/(2-6)= -1/4#

Slope of perpendicular #CF# is #m_2= -1/(-1/4)=4#

Equation of line #CF# passing through #C(7,9)# is

#y-9= 4(x-7) or y-9 = 4x-28 or 4x-y=19 (2)#

Solving equation(1) and (2) we get their intersection point , which

is the orthocenter. Adding equation(1) and (2) we get,

#5x=28 or x = 28/5=5.6 and y= 9-x =9-5.6 =3.4#

Orthocenter of the triangle is at #(5.6,3.4) # [Ans]