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The Altitudes and Orthocenter in a Triangle

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The Altitudes and Orthocenter in a Triangle

Algebra Foundations

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Students learn the definitions of altitude, base of an altitude, foot of an altitude and orthocenter of a triangle, and calculate the coordinates of the orthocenter when given the equations of the lines containing the sides of the triangle or the altitudes.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • After completing this Activity Object, students will be able to:
  • Define altitude.
  • Define the foot of an altitude.
  • Define the orthocenter of a triangle
  • Find the coordinates of the orthocenter of a triangle.

Everything You'll Have Covered

A triangle center is a point in the plane that is, in some sense, similar to the center of a square or a circle. Some triangle centers are the centers of circles related to the given triangle. Examples of triangle centers include the centroid (intersection point of the medians of a triangle), the incenter (intersection point of the internal angle bisectors of a triangle, and the center of the incircle), and the circumcenter (the center of the circle which passes through the three vertices of the triangle). These three centers were known to the Ancient Greeks and can be constructed through the use of a straightedge and compass. All triangle centers are invariant under similarity, which means that they remain the same when the triangle is rotated, reflected, or dilated.

Although the Ancient Greeks discovered several centers of a triangle, they had not constructed a formal definition of the term. In the 1980s, mathematicians noticed that several other points associated with a triangle such as the Fermat point, nine-point center, and symmedian point all shared similar properties with each other and with the incenter, circumcenter, and centroid. From this, the formal definition of a triangle center was created. The definition of triangle center is satisfied by infinitely many objects, but so far, over 3500 triangle centers have been reported and investigated.

  • This Activity Object concerns the orthocenter, another of the triangle centers discovered by the Ancient Greeks. This point is the intersection point of the altitudes of the triangle. We use the following definitions:
  • An altitude is the perpendicular line segment from a vertex to the line containing the opposite side.
  • When an altitude is drawn from a vertex of a triangle, the opposite side is known as its base.
  • The foot of an altitude is the intersection point of the altitude with the line containing the opposite side.
  • The orthocenter is the intersection point of the altitudes of a triangle.
  • It is important to note that the definition of "altitude" refers to the "line containing the opposite side." This means that altitudes do not have to be drawn inside a triangle. In fact:
  • The altitudes are drawn inside the triangle for acute triangles.
  • Two altitudes coincide with the sides of right triangles.
  • Two of the altitudes are drawn outside obtuse triangles.

As a result, the orthocenter of an acute triangle will lie inside the triangle, but will coincide with the vertex opposite the hypotenuse in a right triangle; and will lie outside the triangle in an obtuse triangle.

Tutorial Details

Approximate Time 20 Minutes
Pre-requisite Concepts Students should know the point-slope form of a line, know how to solve a system of equations using the elimination method, and understand the relationship between the slopes of perpendicular lines.
Course Algebra Foundations
Type of Tutorial Skills Application
Key Vocabulary altitude, coordinates of a point, foot of the altitude