What is the orthocenter of a triangle with corners at #(3 ,6 )#, #(4 ,2 )#, and (5 ,7 )#?

Question

What is the orthocenter of a triangle with corners at #(3 ,6  )#, #(4 ,2   )#, and (5   ,7   )#?

Claire Akin · Accepted Answer

Orthocenter of the triangle #color(purple)(O (17/9, 56/9))#

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

Slope of #AD = m_(ad) = - (1/m_(bc) = - (1/5)#

Equation of AD is
#y - 6 = -(1/5) * (x - 3)#

#color(red)(x + 5y = 33)# Eqn (1)

Slope of #AB = m_(AB) = (y_a - y_b) / (x_a - x_b) = (6-2)/(3-4) = -4#

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

Equation of CF is
#y - 7 = (1/4) * (x - 5)#

#color(red)(-x + 4y = 23)# Eqn (2)

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

Solving the two equations,
#x = 17/9, y = 56/9#

Coordinates of orthocenter #color(purple)(O (17/9, 56/9))#

Ella Armstrong · Answer

undefined To find the orthocenter of a triangle, you need to find the point of intersection of the altitudes of the triangle. An altitude is a line segment from a vertex perpendicular to the opposite side.

First, find the slopes of the lines containing the sides of the triangle:

Slope of the line passing through (3, 6) and (4, 2):
m1 = (2 - 6) / (4 - 3) = -4 / 1 = -4

Slope of the line passing through (3, 6) and (5, 7):
m2 = (7 - 6) / (5 - 3) = 1 / 2

Slope of the line passing through (4, 2) and (5, 7):
m3 = (7 - 2) / (5 - 4) = 5 / 1 = 5

The negative reciprocal of the slope of a line is perpendicular to it.

So, the slopes of the altitudes are:
- For the altitude from (3, 6): m1_perpendicular = -1 / m1 = -1 / -4 = 1/4
- For the altitude from (4, 2): m2_perpendicular = -1 / m2 = -1 / (1/2) = -2
- For the altitude from (5, 7): m3_perpendicular = -1 / m3 = -1 / 5

Now, we have the slopes of the altitudes and the coordinates of the vertices. Using the point-slope form of a line, we can find the equations of the altitudes.

1. For the altitude from (3, 6): y - 6 = (1/4)(x - 3)
2. For the altitude from (4, 2): y - 2 = -2(x - 4)
3. For the altitude from (5, 7): y - 7 = (-1/5)(x - 5)

Now, solve these equations to find the points of intersection, which will give you the orthocenter of the triangle.

Allison Allen · Answer

undefined The orthocenter of a triangle is the point of intersection of its three altitudes. To find the orthocenter of a triangle with vertices at (3, 6), (4, 2), and (5, 7):

1. Calculate the slopes of the three sides of the triangle.
2. Find the slopes of the perpendicular lines (altitudes) passing through each vertex.
3. Use the point-slope form to write the equations of these altitudes.
4. Solve the system of equations to find the point of intersection, which is the orthocenter of the triangle.