5.4 Medians and Altitudes

5.4 Medians and Altitudes

Objective of this section:

To identify the properties of medians and altitudes of a triangle

A segment whose endpoints are a vertex and the midpoint of the opposite side.

​

​What is a median?

Theorem 5-8 - Concurrency of Medians Theorem

​What is a CENTROID?

​A centroid is a point of concurrency where the medians of a triangle meet.

The centroid cuts each median so that the long side is two-thirds the length of the entire segment.

Multiple Choice

The CENTROID is the intersection of which special segments?

• Angle Bisectors

• Perpendicular Bisectors

• Medians

• Altitudes

Multiple Choice

The centroid cuts each median into two segments.  The shorter segment is ___________ the length of the entire segment.

• one third
• two thirds
• three fourths
• one half

Multiple Select

If X is a centroid and CW = 15, find CX AND XW.

• CX=5

• XW=5

• CX=10

• XW=10

Fill in the Blank

If P is a centroid and AP = 12, what is the length of AL?

Altitude of a triangle

• A segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side.

• An "Altitude" can be interior or exterior to the triangle.

• Every triangle has 3 altitudes.

Multiple Choice

Is segment AB a median, altitude, neither or both?

• median

• altitude

• neither

• both

Multiple Choice

Is segment AB a median, altitude, neither or both?

• median

• altitude

• neither

• both

Multiple Choice

Is segment AB a median, altitude, neither or both?

• median

• altitude

• neither

• both

Multiple Choice

Is segment AB a median, altitude, neither or both?

• median

• altitude

• neither

• both

​What is an ORTHOCENTER?

​The altitudes are the HEIGHT of the triangle.

​

​An orthocenter is a point of concurrency where the altitudes of a triangle meet.

Multiple Choice

The ORTHOCENTER is the intersection of which special segments?

• Angle Bisectors

• Perpendicular Bisectors

• Medians

• Altitudes

Example 1:

Find the orthocenter using the following points A(1, 1), B(3, 4), and C(6, 1).

Example 2:

Find the orthocenter using the following points A(2,6), B(8,6), and C(6,2).

Draw

Find the orthocenter using the following points A(0,0), B(0,-2), and C(-3,0). Draw a point in YELLOW where the orthocenter should be.

Draw

Find the orthocenter using the following points A(1,2), B(1,6), and C(4,2). Draw a point in YELLOW where the orthocenter should be.

Open Ended

In your own words, explain the difference between a centroid and an orthocenter.

Match

In this lesson and the previous lesson, we have discussed FOUR points of concurrency created by these special segments. Match them below.

• Perpendicular Bisectors

• Angle Bisectors

• Medians

• Altitudes

Circumcenter

Incenter

Centroid

Orthocenter

• Circumcenter

• Incenter

• Centroid

• Orthocenter

Acronym to Help You Remember

Here is an acronym to help you remember what center goes with which type of line.


All Of ---- Altitude → Orthocenter
My Children ---- Median → Centroid
Are Brining In ---- Angle Bisector → Incenter
Peanut Butter Cookies ---- Perpendicular Bisector → Circumcenter

So, we have learned about four special segments/lines in triangles...

Now let's check your understanding....

Multiple Choice

Which center of a triangle is shown in the picture below?

• Circumcenter

• Orthocenter

• Incenter

• Centroid

Multiple Choice

Which center of a triangle is shown in the picture below?

• Circumcenter

• Orthocenter

• Incenter

• Centroid

Multiple Choice

Which center of a triangle is shown in the picture below?

• Circumcenter

• Orthocenter

• Incenter

• Centroid

Multiple Choice

Which center of a triangle is shown in the picture below?

• Circumcenter

• Orthocenter

• Incenter

• Centroid

Poll

Rate your understanding from 1 to 4 (1 being the lowest and 4 being the highest).

• 1

• 2

• 3

• 4