What is the orthocenter of a triangle with corners at #(3 ,1 )#, #(4 ,5 )#, and (2 ,7 )#?

Question

What is the orthocenter of a triangle with corners at #(3   ,1   )#, #(4   ,5   )#, and (2    ,7   )#?

Avery Allen · Accepted Answer

Orthocenter coordinates #color(red)((-1/3), (-7/3)#

Slope of #BC = m_(bc) = (y_b - y_c) / (x_b - x_c) = (5-7)/(4-2) = -1#

Slope of #AD = m_(ad) = - (1/m_(bc) = - (1/-1) = 1#

Equation of AD is
#y - 1 = (1) * (x - 3)#

#color(red)(-x + y = -2)# Eqn (1)

Slope of #AB = m_(AB) = (y_a - y_b) / (x_a - x_b) = (1-5)/(3-4) = 4#

Slope of #CF = m_(CF) = - (1/m_(AB) = - (1/4))#

Equation of CF is
#y - 7 = (4) * (x - 2)#

#color(red)(-4x + y = -1)# Eqn (2)

Solving Eqns (1) & (2), we get the orthocenter #color(purple)(O)# of the triangle

Solving the two equations,
#x = -(1/3), y = -(7/3)#

Coordinates of orthocenter #color(purple)(O (-1/3, -7/3))#

Theodore Abad · Answer

undefined To find the orthocenter of a triangle, you need to first find the intersection point of the altitudes of the triangle. The altitude of a triangle is a perpendicular line from a vertex to the opposite side. Here's how you can find the orthocenter:

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