Question

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

Answer 1

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

Explanation:

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`