Orthocenter of a Triangle
How to solve the orthocenter of a triangle problems: definition, example, and its solution.
The orthocenter of a triangle is the intersecting point
of three heights (altitudes) of a triangle.
To find the orthocenter,
find the intersecting point of two heights (purple, blue).
The purple height is perpendicular to BC,
which is horizontal.
So the purple height is a vertical segment.
And it passes through A(3, 5).
So the purple height's linear equation is x = 3.
To find the linear equation of the blue height,
you need to find the slope of the blue height.
And to find the slope of the blue height,
you need to find the slope of AB: mAB
mAB = 1.
The blue height is perpendicular to AB.
So (blue height's slope, m)⋅(mAB, 1) = -1
You got the blue height's slope: m = -1.
And the blue height passes through C(6, -1).
So the blue height's linear equation is
y - (-1) = -(x - 6).
The blue height's linear equation is y = -x + 5.
The orthocenter is the intersecting point of
x = 3 (purple height) and y = -x + 5 (blue height).
So, by solving this system,
y = 2.
So M(3, 2).