The orthocenter is the intersecting point for all the altitudes of the triangle. One of the traditional questions in the Geometry class is to find the orthocenter of triangle once you have been given the 3 vertices. Hopefully through reading this article, it can help you clear up any doubts you might have about finding the coordinates of the orthocenter.
In our example, we will use the following coordinates as the vertices of the triangle. A (3, 1) B(2, 2) C (3, 5)
Since we are looking for the intersection of the altitudes of the triangle, we only need to find the intersection between 2 of the altitudes as the 3rd altitude will intersect at the exact same point. When you are looking for the altitude of a triangle, it means you are looking for the line from the vertex of a triangle to its opposite side. This line from the vertex will make a perpendicular angle with the opposite line of the triangle.
In order for you to find the equation of a line, you will need to recall your knowledge from Algebra 1. Just as a review, I have also posted 2 articles on this topic. You will need to be able to find the equation of a line, and also the equation of a perpendicular line.
The steps to find the orthocenter are:
- Find the equations of 2 segments of the triangle (for our example we will find the equations for AB, and BC)
- Once you have the equations from step #1, you can find the slope of the corresponding perpendicular lines.
- You will use the slopes you have found from step #2, and the corresponding opposite vertex to find the equations of the 2 lines.
- Once you have the equation of the 2 lines from step #3, you can solve the corresponding x and y, which is the coordinates of the orthocenter.
Step 1: Find equations of the line segments AB and BC.
To find any line segment, you will need to find the slope of the line and then the corresponding y-intercept.
A (3, 1) B(2, 2) C (3, 5)
Slope of AB = (1-2)/(3-2) = -1/1 = -1
y = mx + b (substitute m = -1, x = 3, y = 1)
1 = -1(3) + b
b = 4
Equation of AB: y = -1x + 4
Slope of BC = (2-5)/(2-3) = -3/-1 = 3
y = mx + b (substitute m = 3, x = 2, y = 2)
2 = 3(2) + b
b = -4
Equation of BC: y = 3x - 4
Step 2: Find the slope of the corresponding perpendicular lines
Slope of AB = -1
Slope of perpendicular line to AB: -1*m = -1 -> m = 1
Slope of BC = 3
Slope of perpendicular line to BC: 3*m = -1 -> m = -1/3
Step 3: Find the equation of the perpendicular lines
Slope of perpendicular line to AB: m = 1
We will use the coordinate of the opposite vertex (point C) to find the equation of the line.
y = mx + b (substitute m = 1, x = 3, y = 5)
5 = 1(3) + b
b = 2
Equation of perpendicular line to AB: y = 1x + 2
Slope of perpendicular line to BC: m = -1/3
We will use the coordinate of the opposite vertex (point A) to find the equation of the line.
y = mx + b (substitute m = -1/3, x = 3, y = 1)
1 = -1/3*(3) + b
b = 2
Equation of perpendicular line to AB: y = -1/3x + 2
Step 4: solve 2 perpendicular lines
equation 1: y = 1x + 2
equation 2: y = -1/3x + 2
Solving for x and y:
1x + 2 = -1/3x + 2
4/3x = 0
x = 0
y = 1(0) + 2
y = 2
The coordinates are (0, 2). This is the orthocenter.
If you have any question regarding this type of problems, please feel free to reach out to me or any of the instructors in my center.
Michael Huang
Mathnasium of Glen Rock/Ridgewood
236 Rock Road
Glen Rock, NJ 07452
glenrock@mathnasium.com
Tel: 201-444-8020
