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Video: Orthocenter in Geometry: Definition & Properties



This lesson involves a well-known center of a triangle called the orthocenter. Its definition and properties will be discussed, and an example will be worked showing how to find its location on a graph.

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Orthocenter in Geometry: Definition & Properties

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  • 0:00 The Orthocenter

  • 0:29 Definitions

  • 0:56 Properties and Diagrams

  • 2:04 Special Characteristics

  • 2:41 Example

  • 3:43 Lesson Summary

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Lesson Transcript

Instructor:
Eric Istre

Eric has taught high school mathematics for more than 20 years and has a master’s degree in educational administration.

This lesson involves a well-known center of a triangle called the orthocenter. Its definition and properties will be discussed, and an example will be worked showing how to find its location on a graph.

The Orthocenter

If you had to find the center of a triangle, how would you do it? Well, it would depend on which center you were trying to find. Yes, there is more than one center to a triangle. As a matter of fact, there are many, many centers, but there are four that are most commonly discussed: the circumcenter, the incenter, the centroid, and the orthocenter. The center we are going to discuss in this lesson is the orthocenter.

Definitions

Let’s begin with a basic definition of the orthocenter. The orthocenter is the point of concurrency of the three altitudes of a triangle. Since a triangle has three vertices, it also has three altitudes. An altitude is defined as a perpendicular segment drawn from the vertex of a triangle to the line containing the opposite side. A point of concurrency is the point of intersection of three or more lines.

Properties and Diagrams

There are three types of triangles with regard to the angles: acute, right, and obtuse. When we are discussing the orthocenter of a triangle, the type of triangle will have an effect on where the orthocenter will be located. Take a look at the following diagrams. Typically, we would expect the center of a triangle to be inside of it. Indeed, the orthocenter will always be inside of an acute triangle.

Triangle 1

However, the orthocenter is on a right triangle (specifically at the vertex of the right angle). The only way to draw a segment from point B to point C is to travel along one of the sides of the triangle. The same is true when going from point A to point C.

Triangle 2

Finally, the orthocenter will be outside of an obtuse triangle (specifically opposite the longest side). The only way to draw a segment from point B to the opposite side and it also be perpendicular is to extend side AC. The same is true when drawing the segment from point A to side BC. This forces the point of concurrency to be outside the triangle.

Triangle 3

Special Characteristics

It is worth mentioning that the orthocenter has a few important connections with the other common centers of a triangle. It is always collinear, lying on the same straight line, with the circumcenter and the centroid. This line is named the Euler line. The centroid will always be between the orthocenter and the circumcenter, and the distance from the centroid to the orthocenter will always be twice that of the distance from the centroid to the circumcenter. Also, if the triangle is equilateral, all four of the common centers will be at the exact same location.

Example

We have a triangle with vertices A at ( -2 , 0 ) , B at ( 4 , 0 ) , and C at ( 3 , 5 ). Find the orthocenter.

You can see in this diagram that the triangle is acute. You should expect the orthocenter to be located inside the triangle.

Orthocenter example

The slope of side AC = 5/5, which is the same as 1/1. Therefore, the slope of the altitude from point B to side AC would be -1/1.

The slope of side BC = -5/1; therefore, the slope of the altitude from point A to side BC would be 1/5.

The altitude from point C to side AB must be vertical since side AB is on the x-axis, which makes it horizontal.

The three altitudes cross at the point ( 3 , 1 ).

Lesson Summary

The orthocenter is one of the four most common centers of a triangle. It is located at the point where the triangle’s three altitudes intersect called a point of concurrency. The orthocenter is located inside an acute triangle, on a right triangle, and outside an obtuse triangle. Finding it on a graph requires calculating the slopes of the triangle sides. Then, use the opposite reciprocal of each slope to graph each altitude.


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  4. Orthocenter Coordinates in a Triangle — Practice Geometry Questions

Orthocenter Coordinates in a Triangle — Practice Geometry Questions

Related Book

Geometry: 1,001 Practice Problems For Dummies (+ Free Online Practice)

By Allen Ma, Amber Kuang

To find the orthocenter of a triangle, you need to find the point where the three altitudes of the triangle intersect. In the following practice questions, you apply the point-slope and altitude formulas to do so.

Practice questions

Use your knowledge of the orthocenter of a triangle to solve the following problems.

  1. The coordinates of

    image0.png

    are A (0, 2), B (–2, 6), and C (4, 0). Find the coordinates of the orthocenter of this triangle.

  2. The coordinates of

    image1.png

    are A (0, 0), N (6, 0), and D (–2, 8). Find the coordinates ofthe orthocenter of this triangle.

Answers and explanations

  1. (–8, –6)

    The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. An altitude of a triangle is perpendicular to the opposite side. Because perpendicular lines have negative reciprocal slopes, you need to know the slope of the opposite side. Here’s the slope of

    image2.png

    This means that the slope of the altitude to

    image3.png

    needs to be 1.

    The point-slope formula of a line is y y1 = m (x x1), where m is the slope and (x1, y1) are the coordinates of a point on the line. To find the altitude formed when you connect Point A to

    image4.png

    plug in m = –1 and the coordinates of Point A, (0, 2):

    image5.png

    The slope of

    image6.png

    is

    image7.png

    This means that the slope of the altitude to

    image8.png

    The altitude formed when you connect Point C, (4, 0), to

    image9.png

    is

    image10.png

    To find the orthocenter, you need to find where these two altitudes intersect. Set them equal and solve for x:

    image11.png

    Now plug the x value into one of the altitude formulas and solve for y:

    image12.png

    Therefore, the altitudes cross at (–8, –6).

  2. (–2, –2)

    The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. An altitude of a triangle is perpendicular to the opposite side. Because perpendicular lines have negative reciprocal slopes, you need to know the slope of the opposite side. Here’s the slope of

    image13.png

    This means that the slope of the altitude to

    image14.png

    needs to be 1.

    The point-slope formula of a line is y y1 = m (x x1), where m is the slope and (x1, y1) are the coordinates of a point on the line. To find the altitude formed when you connect Point A to

    image15.png

    plug in m = 1 and the coordinates of A, (0, 0):

    image16.png

    Now find the equation for the altitude to

    image17.png

    The slope of

    image18.png

    This means that the slope of the altitude to

    image19.png

    The altitude formed when you connect Point N, (6, 0), to

    image20.png

    To find the orthocenter, you need to find where the two altitudes intersect. Set them equal and solve for x:

    image21.png

    Now plug the x value into one of the altitude formulas and solve for y:

    y = x

    y = –2

    Therefore, the altitudes cross at (–2, –2).

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