The ORTHOCENTER (O) of a triangle is the common intersection
of the three sides containing the altitudes. An altitude is a perpendicular
segment from a vertex to the line of the opposite side.
In the scalene triangle above, the orthocenter is inside of the
triangle where the altitudes meet.
Is this the case for all triangles?
To explore the location of the orthocenter for various shapes of triangles, click here.
Did the location of the orthocenter move?
Intuitively, the answer to this question should be yes. The orthocenter is the intersection of the altitudes. Hence,when you construct new triangles of different shapes and sizes, the location of the orthocenter will also move and change position.
Now that you have explored various shapes of triangles and its effect on the location of the orthocenter, consider several of the distinct cases.
Consider what happens with the position of the orthocenter when the triangle is right, obtuse, and equilateral.
First, intuitively what do you think will happen to the location of the orthocenter when you have one of these triangles?
Now consider each case to see if your assumptions are correct.
LOCATION OF ORTHOCENTER IN A RIGHT TRIANGLE
When the triangle is right, the orthocenter is the vertex of the triangle at the right angle. Intuitively this makes sense because the orthocenter is where the altitudes intersect. Hence, in a right triangle, the vertex of the right angle is where you would expect the altitudes to meet, at 90 degrees, where the legs of the right triangle are perpendicular.
LOCATION OF ORTHOCENTER IN AN OBTUSE TRIANGLE
When an obtuse triangle is constructed and the altitudes are drawn, the intersection of the altitudes is outside of the triangle. Therefore, the orthocenter moves completely out of the triangle where the altitudes meet. Students often have difficulty with obtuse triangles and their respective altitudes and location of the altitudes. It is important to give students many opportunities dealing with obtuse triangles not just acute triangles.
LOCATION OF ORTHOCENTER IN AN EQUILATERAL TRIANGLE
In an equilateral triangle, the lengths of the segments of the sides of the triangle are all equal. Thus, the distance from the orthocenter to each side of the triangle are equal. In addition, the orthocenter lies on the perpendicular bisector of each side of the triangle. Another conclusion is the distance from the orthocenter to each vertex of the triangle are equal.