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.
After completing this tutorial, you will be able to complete the following:
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.
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.
|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.|
|Type of Tutorial||Skills Application|
|Key Vocabulary||altitude, coordinates of a point, foot of the altitude|