Orthocenter of a triangle
An altitude of a triangle (ha, hb y hc) is a perpendicular line segment from a vertex to the opposite side. This line containing the opposite side is called the extended base of the altitude.
Altitude can also be understood as the distance between the base and the vertex.
Where is the Orthocenter of a Triangle Located?
- If it’s an obtuse triangle the orthocenter is located outside the triangle (as we see in the picture above).
- If it’s an acute triangle the orthocenter is located inside the triangle.
- If it’s a right triangle the orthocenter lies on the vertex of the right angle.
In an equilateral triangle all three centers are in the same place.
The relative distances between the triangle centers remain constant.
Distances between centers:
Find the coordinates of the orthocenter H of a triangle ABC with the vertex coordinates A (3, 5), B (4, -1), and C (-4, 1).
In order to find the orthocenter we have to solve the equations for the altitudes ha (starting from vertex A) and hb (from vertex B) and see where is located the intersection point (H) of both altitudes.
STEP 1: Find equation for ha:
Firstly we will find the equation of the line that passes through side BC, which is the opposite of vertex A. This equation is obtained knowing that it passes through points B (4, -1) and C (-4, 1). The general equation of the line that passes through two known points is:
The equation of the line that contains side BC and its slope m will be:
The slope of the line that contains the altitude ha, being perpendicular to the side BC, is the inverse and of the opposite sign to the slope of the line that contains the side. So, we have that:
Knowing that the height ha passes through the vertex A (3, 5), we can obtain the equation of its line. The equation of a line, knowing a point and its slope (inverse slope and opposite sign to that found for BC, that is, mp = 4):
So, the equation for ha is y -4x +7 = 0.
STEP 2: Find equation for hb:
Now we proceed in the same way to find the equation of the line that contains the height hb, that is, to which, starting from vertex B, is perpendicular to side AC.
First, the slope of the line AC:
And, now, the equation of the line that contains the height hb, that is, the one that starts from vertex B and is perpendicular to side AC. Therefore, its slope mp will be –7/4.
Then, the equation of hb is 4y +7x -24 = 0.
We solve this system of two first degree equations with two unknowns. Its roots will be the coordinates of its intersection, that is, the location of the orthocenter H of the triangle ABC.
Solving this system of two first-degree equations we have that:
Therefore, the altitudes cross at (2.26, 2.04).
And the coordinates of the orthocenter are H (2.26, 2.04).
AUTHOR: Bernat Requena Serra