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 = (4,9) , B(3,7) , C(1,1) # . 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= (1-7)/(1-3)= 3#
Slope of perpendicular #AD# is #m_2= -1/3 (m_1*m_2=-1) #
Equation of line #AD# passing through #A(4,9)# is #y-9= -1/3(x-4)# or
#y-9 = -1/3 x+4/3 or y +1/3x = 9+4/3 or y +1/3x = 31/3 (1)#
Slope of #AB# is #m_1= (7-9)/(3-4)= =2#
Slope of perpendicular #CF# is #m_2= -1/2 (m_1*m_2=-1) #
Equation of line #CF# passing through #C(1,1)# is #y-1= -1/2(x-1)# or
#y-1 = -1/2 x+1/2 or y +1/2x = 1+1/2 or y +1/2x = 3/2 (2)#
Solving equation(1) and (2) we get their intersection point , which is the orthocenter.
#y +1/3x = 31/3 (1)#
#y +1/2x = 3/2 (2) # Subtracting (2) from (1) we get,
#-1/6x = (31/3-3/2) = 53/6 or x = - 53/cancel6*cancel6 or x=-53#
Putting #x= -53# in equation (2) we get #y-53/2 =3/2 or y=53/2+3/2 or 56/2=28 :. x= -53 ,y = 28#
Orthocenter of the triangle is at #( -53,28) # [Ans]
