Triangle: Orthocenter

How to find the orthocenter of a triangle: definition, 1 example, and its solution.

Definition

Definition

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

Example

Example

Solution

The orthocenter of a triangle
can be found by
finding the intersecting point
of these two heights.

So, find the linear equations
that show these two heights.

First, find this height.

The y values of B and C are both -1.
So BC is a horizontal side.

BC and the height is perpendicular.
So the height is vertical.

The x value of A is 3.
So the linear equation that shows the height is
x = 3.

Next, find the other height.

The linear equation that shows the height
passes through C(6, -1).
But you don't know the slope.

The height is perpendicular to AB.

So, to find the slope of the height,
find the slope of AB.

A(3, 5)
B(-3, -1)

Then the slope of AB is
mAB = [5 - (-1)]/[3 - (-3)].

5 - (-1) = 5 + 1

3 - (-3) = 3 + 3

5 + 1 = 6

3 + 3 = 6

6/6 = 1

So the slope of AB is
mAB = 1.

Set the slope of the height m.
mAB = 1

The height and AB are perpendicular.

So m⋅1 = -1.

Perpendicular Line Equation

So m = -1.

So the slope of the height is
m = -1.

m = -1
The height passes through C(6, -1).

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

-1(x - 6) = -x + 6

+6 - 1 = +5

So y = -x + 5.

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

The linear equations that show the slopes are
[x = 3] and [y = -x + 5].

Find the intersecting point,
which is the orthocenter,
by solving this system of linear equations.

Put x = 3
into y = -x + 5.

Then y = -3 + 5.

Substitution Method

-3 + 5 = 2

x = 3
y = 2

So the intersecting point is
M(3, 2).

So the orthocenter is M(3, 2).