Question

What is the orthocenter of a triangle with corners at `(2 ,6 )`, `(9 ,1 )`, and (5 ,3 )#?

Answer 1

The Orthocenter is `(-10,-18)`

Explanation:

The Orthocenter of a triangle is the point of intersection of the 3 altitudes of the triangle.

The slope of the line segment from point `(2,6)` to `(9,1)` is:

`m_1 = (1-6)/(9-2)`

`m_1 = -5/7`

The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:

`p_1 = -1/m_1`

`p_1 = -1/(-5/7)`

`p_1 = 7/5`

The altitude must pass through point `(5,3)`

We can use the point-slope form for the equation of a line to write the equation for the altitude:

`y = 7/5(x-5)+3`

Simplify a bit:

`y = 7/5x-4" [1]"`

The slope of the line segment from point `(2,6)` to `(5,3)` is:

`m_2 = (3-6)/(5-2)`

`m_2 = -3/3`

`m_2 = -1`

The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:

`p_2 = -1/m_2`

`p_2 = -1/(-1)`

`p_2 = 1`

The altitude must pass through point `(9,1)`

We can use the point-slope form for the equation of a line to write the equation for the altitude:

`y = 1(x-9)+1`

Simplify a bit:

`y = x-8" [2]"`

We could repeat this process for the third altitude but we have already enough information to determine the intersection point.

Set the right side of equation [1] equal to the right side of equation [2]:

`7/5x-4 = x-8`

Solve for the x coordinate of intersection:

`2/5x =-4`

`x = -10`

To find the value of y, substitute -10 for x into equation [2]:

`y = -10 - 8`

`y = -18`

The Orthocenter is `(-10,-18)`