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

Question

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

Sadie Adams · Accepted Answer

Coordinates of Orthocenter #color(blue)(O (56/11, 20/11))#

Orthocenter is the concurring point of the three altitudes of a triangle and represented by ‘O’

Slope of BC #= m_a = (6-2) / (3-6) = -(4/3)#

#Slope of AD = -(1/m_a) = (3/4)#

Equation of AD is

#y - 1 = (3/4) (x - 4)#

#4y - 3x = -8# Eqn (1)

Slope of AB #= m_c = (2 - 1) / 6-4) = (1/2)#

Slope of CF = -(1/m_c) = -2#

Equation of CF is

#y - 6 = -2 (x - 3)#

#y + 2x = 12# Eqn (2)

Solving Eqns (1), (2)

#x = 56/11 , y = 20/11#

we get the coordinates of Orthocenter #color(blue)(O (56/11, 20/11))#

Verification

Slope #m_b = (6-1) / (3-4) = -5#

Slope of BE = -(1/m_c) = 1/5#

Equation of altitude BE is

#y - 2 = (1/5) (x - 6)#

#5y - 10 = x - 6#

#5y - x = 4# Eqn (3)

Solving equations (2) , (3),

Coordinates of #color(blue)(O (56/11, 20/11)#

Theodore Abad · Answer

undefined To find the orthocenter of a triangle with vertices at (4, 1), (6, 2), and (3, 6), follow these steps:

1. Determine the slopes of the lines passing through each pair of vertices.
2. Calculate the slopes of the perpendicular bisectors of each side of the triangle.
3. Find the point of intersection of these perpendicular bisectors.
4. This point is the orthocenter of the triangle.

Performing these calculations will give you the coordinates of the orthocenter.