Kuzas, Richmond, Tocchi – Team 6 – Orthocenter
Altitudes and the Orthocenter of a Triangle
Lesson Summary:
Students will use software to explore the properties of the altitudes of a triangle, and the point
where the altitudes meet in a triangle. Students will explore obtuse, right, and acute triangles.
Key Words:
altitudes, heights, orthocenter
Background Knowledge:
Students should be familiar with Geometry software, as well as with the classification of
triangles based upon their angles.
NCTM Standards Addressed:
Strand 4 Geometry; Standard 7 – Geometry from a synthetic perspective
Learning Objectives:
1. To understand the concept of the altitude of a vertex of a triangle as the perpendicular
line from the vertex to the line containing the opposite side of the triangle.
2. To understand the difference between altitude and height of a triangle.
3. To be able to understand orthocenter and its characteristics
Materials:
Cabri II
Suggested Procedure:
Split students into groups of two or three. Have students complete the worksheets.
Assessment:
Completed worksheets should serve as assessment.
Kuzas, Richmond, Tocchi – Team 6 – Orthocenter
Activity I. Altitudes of a Triangle
Activity goals: To construct and define an altitude of a triangle. To understand the different
locations of the altitude based upon the type of triangle.
1. Draw a triangle and label the vertices A, B, and C.
(use the triangle and label tools)
2. Draw a line t through the points B and C.
(use the line and label tools)
3. Draw a perpendicular line h from vertex A to line t (make sure to point to the line outside
the triangle).
(use the perpendicular line tool)
4. Draw the point of intersection of lines h and t and label it X.
(use the point and label tools)
5. Draw the segment AX .
(use the segment tool)
The segment AX is called the altitude of the ABC from vertex A. The length of the
segment AX is called the height of the triangle from vertex A.
6. Make the lines h and t dotted. Draw segment BC .
(use the attribute toolbox)
7. Grab and move around vertex A. What do you notice about the location of the altitude?
The location of the altitude moves with vertex A. It remains perpendicular to segment BC
8. For what type of triangles does the altitude fall outside the triangle?
Obtuse triangles.
9. For what type of triangles does the altitude fall on a side of the triangle?
Right triangles.
10. For what type of triangles does the altitude fall inside the triangle?
Acute triangles.
Kuzas, Richmond, Tocchi – Team 6 – Orthocenter
Activity II. Orthocenter of a Triangle
Lab Goals: Discover the properties of the orthocenter. The orthocenter is the intersection of the
altitudes of a triangle. Pay close attention to the characteristics of the orthocenter in obtuse,
acute, and right triangles.
1. Draw triangle ABC .
(use triangle tool)
2. Draw the lines AC , AB , and BC containing the sides of the triangle.
(use line tool)
3. Draw perpendicular lines from each vertex to the lines AC , AB , and BC containing
the opposite side. Label the points of intersection X, Y, and Z.
(use perpendicular tool)
4. Draw the altitudes (the segments AZ , BY , and CX ). (use point intersection and label tools)
5. Label the point of intersection of the altitudes M.
6. Drag any of the three vertices A, B, and C to different positions. What do you notice
about the measure of the angles X, Y, and Z?
(use angle measure tool)
o
The measures of angles X, Y, and Z are always 90 .
7. Is there ever a time when there is not an intersection of all three perpendicular lines?
The only time I found that the three perpendicular lines didn’t cross was when I made the
triangle a line. Therefore, they always intersect.
The point of intersection of the altitudes of a triangle is called the orthocenter of the
triangle.
8. Drag a vertex so that the triangle is an acute triangle. What do you notice about the
location of the orthocenter?
The orthocenter is located on the inside of the triangle.
9. Drag a vertex so that the triangle is a right triangle. Where is the orthocenter located
now?
The orthocenter is located on the vertex of the angle that is 90o.
10. Drag a vertex so that the triangle is an obtuse triangle. What do you notice about the
orthocenter?
The orthocenter is located outside the triangle by the angle that was made obtuse.
Kuzas, Richmond, Tocchi – Team 6 – Orthocenter
11. What can you conclude about the orthocenter of a triangle?
It can be located on a vertex, inside or outside of the triangle. The orthocenter is the
intersection of the altitude lines of each side of the triangle.
Extension
I.
Create a macro “Orthocenter,” to find the orthocenter of a triangle given by its three
vertices.
II.
1. Draw a new triangle ABC .
2. Draw from each vertex, a parallel line to the opposite side of the triangle.
3. Label the new points of intersection X, Y, and Z respectively to create the new
triangle XYZ .
4. Construct the orthocenter of ABC using the macro “Orthocenter.”
5. Measure segments XA , and AY . What is point A with respect to XY ?
Point A is the midpoint of the line.
Kuzas, Richmond, Tocchi – Team 6 – Orthocenter
6. What role does the altitude AO play with respect to the side XY of triangle
XYZ ?
(Hint: A line  to another line t is  to any line parallel to t.)
AO is perpendicular to XY because it is perpendicular to BC which is parallel
to XY .
7. It follows that the orthocenter of ABC becomes what important center of
XYZ ?
The orthocenter of ABC is the circumcenter of XYZ .
Kuzas, Richmond, Tocchi – Team 6 – Orthocenter
Journal Activity
Finding the Orthocenter of a Triangle
1. List all definitions and properties that you have learned in this activity.
Orthocenter- The point of intersection of the three altitudes of a triangle. The orthocenter
can be located inside or outside of the triangle.
2. Can you think of any application of the orthocenter of a triangle?
No
3. Can you relate this topic/concept with other(s) previously studied? Explain your answer.
This is the only intersection point of the triangle we have studied for which I cant identify
a use for. Will be interested to see how this topic is discussed in class.