Exploring Altitudes and the Orthocenter

 

by:

 

Brandt Hacker

 

The orthocenter is the point of concurrency of the three altitudes of a triangle.  To understand what this means, we must first determine what an altitude is.  An altitude is a line that passes through a vertex of a triangle and that is perpendicular to the line that contains the opposite side of said vertex.  Below are two examples of altitudes, one in which the altitude is perpendicular to the opposite side, and one in which the altitude is perpendicular to the line that contains the opposite side.

 

Here we have an altitude that goes through vertex B and is perpendicular to side AC.  Below we will look at the second scenario, an altitude that goes through a vertex and that is perpendicular to the line that contains the opposite side, rather than the side itself.

 

In the image shown above, we see that there is no such line that passes through vertex B and is perpendicular to side AC.  In this instance, the altitude goes through vertex B and is perpendicular to the dotted line that contains side AC.  The altitude is pictured in red.  What type of triangle created the situation in which the altitude could not intersect the triangle’s side?

 

Now that we have a better grasp on what an altitude is, we are ready to begin discussing the orthocenter.  Every triangle has three altitudes, one that runs through each vertex.  When all three altitudes are drawn on the same triangle, they intersect at exactly one point, the orthocenter.  For a tool that allows you to test this, click HERE.

 

We will now examine the orthocenter of three different triangles:  acute, right, and obtuse.  If you explored the tool that creates the three altitudes, you may have already been able to make some conjectures about how the orthocenter changes.

 

First we will look at an acute triangle.  For any acute triangle, the three altitudes of the triangle intersect a vertex and opposite side of the triangle.  Therefore, for an acute triangle the orthocenter will always be on the interior of the triangle.  One such example is show below.

 

Next we will look at an obtuse triangle.  For any obtuse triangle two of the altitudes will intersect the line containing the opposite side, not the side itself.  This causes the orthocenter to lie outside the triangle.  One such example is shown below.

 

 

Lastly we will examine the right triangle.  Due to the fact that two of the legs of the right triangle form a right angle and pass through vertices of the triangle, the orthocenter of a right triangle will always lie on the vertex of the triangle’s right angle.  We see this in the image below.

 

Using what we’ve explored about the orthocenter of various triangles, try and determine the following.

 

  1. What type of triangle would have an altitude that runs through a vertex and intercepts the midpoint of the opposite side?

 

  1. What type of triangle would have an altitude that is also a median?  Three altitudes that are all medians?

 

Explore these questions using the link to the tool that creates the altitudes, found HERE.

 

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