What is the orthocenter of a triangle with corners at #(9 ,7 )#, #(2 ,9 )#, and (5 ,4 )#?

Question

What is the orthocenter of a triangle with corners at #(9  ,7   )#, #(2   ,9   )#, and (5    ,4   )#?

Noah Atkinson · Accepted Answer

The orthocenter G is point #(x = 151/29 ,y = 137/29)#

The figure below depicts the given triangle and the associated heights (green lines) from each corner.The orthocenter of the triangle is point G.

The orthocentre of a triangle is the point where the three altitudes meet.
You need to find the equation of the perpendicular lines that pass through two at least of the triangle vertices.

First determine the equation of each of the sides of the triangle:

From A(9,7) and B(2,9) the equation is

#2 x+7 y-67 = 0#

From B(2,9) and C(5,4) the equation is

#5 x+3 y-37 = 0#

From C(5,4) and A(9,7) the equation is

#-3 x+4 y-1 = 0#

Second, you must determine the equations of the perpendicular lines that pass through each vertex:

For AB through C we have that

#y = (7 (x-5))/2+4#

For AC through B we have that

#y = 9-(4 (x-2))/3#

Now point G is the intersection of the heights hence we have to solve the system of two equations

#y = (7 (x-5))/2+4# and #y = 9-(4 (x-2))/3#

Hence the solution gives the coordinates of the orthocenter G

#x = 151/29 ,y = 137/29#

Mason Adams · Answer

undefined To find the orthocenter of a triangle, you need to first find the altitude of each side of the triangle. Then, the point where all three altitudes intersect is the orthocenter.

Altitude of a side of a triangle is the perpendicular line from a vertex to the opposite side.

To find the altitude from a vertex to the opposite side, you need to:

1. Find the slope of the line containing the side of the triangle.
2. Use the negative reciprocal of that slope to find the slope of the altitude.
3. Use the point-slope form of a line to find the equation of the altitude.
4. Find the intersection point of the altitudes.

After finding the equations of all three altitudes, you can solve them simultaneously to find the orthocenter.

Alternatively, you can use geometric methods to find the orthocenter by constructing the perpendiculars from the vertices to the opposite sides.

Performing these calculations for the given triangle with vertices at (9,7), (2,9), and (5,4), will yield the coordinates of the orthocenter.