Orthocenter of a Triangle

Orthocenter of a Triangle

How to find the orthocenter of a triangle: definition, example, and its solution.

Definition

The orthocenter of a triangle is the intersecting point of three heights (altitudes) of a triangle.

The orthocenter of a triangle
is the intersecting point
of three heights (altitudes) of a triangle.

Example

Find the coordinates of the centriod of triangle ABC. A(3, 5), B(-3, -1), C(6, -1).

To find the orthocenter M,
find the intersection of these two heights.

So find the linear equations of these two heights.

First, find the linear equation of the first height.

The endpoints of BC are (-3, -1) and (6, -1).
Their y values are -1.
So BC is a horizontal side.

BC and the height are perpendicular.

So the first height is a vertical segment.

A(3, 5) is on the blue height.

So the linear equation of the first height is
x = 3.

Slope of a line - No slope

Next, find the linear equation of the second height.

C(6, -1) is on the height.
The slope is not given.

But you can find the slope of the height
by using the slope of AB,
because AB and the height are perpendicular.
(You'll see the details below.)

So find the slope of AB: mAB.

Its endpoints are (-3, -1) and (3, 5).

So mAB = [5 - (-1)]/[3 - (-3)]
= 1.

Slope of a line

mAB = 1
Set the slope of the height m.

AB and the height are perpendicular.

So 1⋅m = -1.
So m = -1.

Perpendicular lines

The slope of the height, m, is -1.
And C(6, -1) is on the height.

Then the linear equation in point-slope form is
y = -(x - 6) - 1.

Point-slope form

-(x - 6) = -x + 6

6 - 1 = 5

So the linear equation of the second height is
y = -x + 5.

The linear equation of the heights are
[x = 3] and [y = -x + 5].

The intersection is the orthocenter M.

M is on [x = 3].
So the x value of M is 3.

To find the y value of M,
put [x = 3] into [y = -x + 5].

Then y = 2.

Substitution method

x = 3
y = 2

So the orthocenter M is (3, 2).