Title: READ: Altitude Reading
Course: Geometry (A)

Altitudes in Triangles

Learning Objectives

- Construct the altitude of a triangle.
- Apply the Concurrency of Altitudes Theorem to identify the point of concurrency of the altitudes of the triangle (the orthocenter).
- Use the Concurrency of Altitudes Theorem to solve problems involving the orthocenter of triangles.

Introduction

In this lesson we will conclude our discussions about special line segments associated with triangles by examining altitudes of triangles. We will learn how to find the location of a point within the triangle that involves the altitudes.

Definition of Altitude of a Triangle

An altitude of a triangle is the line segment from a vertex perpendicular to the opposite side. Here is an example that shows the altitude from vertex in an acute triangle.

We need to be careful with altitudes because they do not always lie inside the triangle. For example, if the triangle is obtuse, then we can easily see how an altitude would lie outside of the triangle. Suppose that we wished to construct the altitude from vertex in the following obtuse triangle:

In order to do this, we must extend side as follows:

Will the remaining altitudes for (those from vertices and ) lie inside or outside of the triangle?

Answer: The altitude from vertex will lie inside the triangle; the altitude from vertex will lie outside the triangle.

As was true with perpendicular bisectors (which intersect at the circumcenter), and angle bisectors (which intersect at the incenter), and medians (which intersect at the centroid), we can state a theorem about the altitudes of a triangle.

Concurrency of Triangle Altitudes Theorem: The altitudes of a triangle will intersect in a point. We call this point the orthocenter of the triangle.

Rather than prove the theorem, we will demonstrate it for the three types of triangles (acute, obtuse, and right) and then illustrate some applications of the theorem.

Acute Triangles

The orthocenter lies within the triangle.

Obtuse Triangles

The orthocenter lies outside of the triangle.

Right Triangles

The legs of the triangle are altitudes. The orthocenter lies at the vertex of the right angle of the triangle.

Lesson Summary

In this lesson we:

- Defined the orthocenter of a triangle.
- Stated the Concurrency of Altitudes Theorem.
- Solved problems using the Concurrency of Altitudes Theorem.

Points to Consider

Remember that the altitude of a triangle is also its height and can be used to find the area of the triangle. The altitude is the shortest distance from a vertex to the opposite side.

The following questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

1. In our lesson we looked at the special case of an isosceles triangle and determined relationships about among the incenter, circumcenter, centroid, and orthocenter. Explore the case of an equilateral triangle and see which (if any) relationships hold.
2. Perform the same exploration for an acute triangle. What can you conclude?
3. Perform the same exploration for an obtuse triangle. What can you conclude?
4. Perform the same exploration for a right triangle. What can you conclude?
5. What can you conclude about the four points for the general case of ?
6. In 3 you found that three of the four points were collinear. The segment joining these three points define the Euler segment. Replicate the exploration of the general triangle case and measure the lengths of the Euler segment and the sub-segments. Drag your drawing so that you can investigate potential relationships for several different triangles. What can you conclude about the lengths?
7. (Found in Exploring Geometry, 1999, Key Curriculum Press) Construct a triangle and find the Euler segment. Construct a circle centered at the midpoint of the Euler segment and passing through the midpoint of one of the sides of the triangle.
This circle is called the nine-point circle. The midpoint it passes through is one of the nine points. What are the other eight?
8. Consider with, , altitudes of the triangles as indicated.
Prove: .
9. Consider isosceles triangle with , and .
Prove: .

Review Answers

1. All four points are the same.
2. The four points all lie inside the triangle.
3. The four points all lie outside the triangle.
4. The orthocenter lies on the vertex of the right angle and the circumcenter lies on the midpoint of the hypotenuse.
5. The orthocenter, the circumcenter, and the centroid are always collinear.
  1. The circumcenter and the orthocenter are the endpoints of the Euler segment.
  2. The distance from the orthocenter to the centroid is twice the distance from the centroid to the circumcenter.
7. Three of the points are the midpoints of the triangle’s sides. Three other points are the points where the altitudes intersect the opposite sides of the triangle. The last three points are the midpoints of the segments connecting the orthocenter with each vertex.
8. The congruence can be proven by showing the congruence of triangles and . This can be done by applying postulate AAS to the two triangles.
9. The proof can be completed by using the AAS postulate to show that triangles and are congruent. The conclusion follows from CPCTC.