Altitudes in Triangles
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C ONCEPT
Concept 1. Altitudes in Triangles
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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 A 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 A in the following obtuse triangle:
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In order to do this, we must extend side CB as follows:
Will the remaining altitudes for 4ABC (those from vertices B and C ) lie inside or outside of the triangle?
Answer: The altitude from vertex B will lie inside the triangle; the altitude from vertex C 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
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Concept 1. Altitudes in 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.
Even with these three cases, we may still encounter special triangles that exhibit interesting properties.
Example 1
Use a piece of Patty Paper (500 ×500 tracing paper), or any square piece of paper to explore orthocenters of isosceles
4ABC. Note: Patty Paper may be purchased in bulk from many Internet sites.
Determine any relationships between the location of the orthocenter and the location of the incenter, circumcenter,
and centroid.
First let’s recall that you can construct an isosceles triangle with Patty Paper as follows:
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1. Draw line segment AB.
2. Fold point A onto point B to find the fold line.
3. Locate point C anywhere on the fold line and connect point C to points A and B. (Hint: Locate point C as far
away from A and B as possible so that you end up with a good-sized triangle.). Trace three copies of 4ABC onto
Patty Paper (so that you end up with four sheets of paper, each showing 4ABC).
4. For one of the sheets, fold the paper to locate the median, angle bisector, and perpendicular bisector relative to
the vertex angle at point C. What do you observe? (Answer: They are the same line segment.) Fold to find another
bisector and locate the intersection of the two lines, the incenter.
5. For the second sheet, locate the circumcenter of 4ABC.
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Concept 1. Altitudes in Triangles
6. For the third sheet, locate the centroid of 4ABC.
7. For the third sheet, locate the orthocenter of 4ABC.
8. Trace the location of the circumcenter, centroid, and orthocenter onto the original triangle. What do you observe
about the four points? (Answer: The incenter, orthocenter, circumcenter, and centroids are collinear and lie on the
median from the vertex angle.)
Do you think that the four points will be collinear for all other kinds of triangles? The answer is pretty interesting!
In our homework we will construct the four points for a more general case.
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.
Examined the special case of an isosceles triangle and determined relationships about among the incenter,
circumcenter, centroid, and orthocenter.
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.
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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 4ABC 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 4ABC?
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 4ABC∼
=4DEF with, AP, DO altitudes of the triangles as indicated.
Prove: AP ∼
= DO.
9. Consider isosceles triangle 4ABC with AB ∼
= AC, and BD ⊥ AC,CE ⊥ AB. Prove: BD ∼
= CE.
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.
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Concept 1. Altitudes in Triangles
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.
a. The circumcenter and the orthocenter are the endpoints of the Euler segment.
b. The distance from the orthocenter to the centroid is twice the distance from the centroid to the circumcenter.
6. 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.
7. The congruence can be proven by showing the congruence of triangles 4APB and 4DOE. This can be done
by applying postulate AAS to the two triangles.
8. The proof can be completed by using the AAS postulate to show that triangles 4CEB and 4BDC are
congruent. The conclusion follows from CPCTC.
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