Assignment #4

Medians and Centroids

by

Vicki Tarleton

The following is an exploration of medians of a triangle that can be used with geometry students.

Definition: The median of a triangle is the segment from a vertex to the midpoint of the opposite side.

Instructions for Geometer's Sketch Pad Exploration

Construct a Median and explore

Construct a triangle, ABC in GSP. Then construct the median BR.

How do you think the area of triangle ABC compares to the area of triangle ABC? to the area of triangle ABR? Make a conjecture.

Now find out if your conjecture is correct.

Select points A, B, and C. Under the "construct" menu, choose polygon interior. Next, under the "measure" menu choose area.

Now select points R, B, and C and complete the same steps as above.

Select points A, B, and R and complete the steps again.

Now measure angle ACB. Move point B (make sure to notice area while moving) so as to make a right triangle. Continue to move point B to make an obtuse triangle.

Does you conjecture hold for all types of triangles?

Construct all three Medians and explore

Construct the other two medians of triangle ACB; median CS and median AT.

Do you notice anything about the intersection of the three medians?

Since they all seem to intersect at one point, let's see if this holds when the triangle is changed.

Drag point B to various locations.

Do the three medians keep one intersection point when you drag point B?

Definition: The three medians do have a common intersection point called the centroid of the triangle. The letter G is used to represent the centroid of a triangle.

Characteristics of the Centroid

Once again, drag point B so as to have an acute, right, and obtuse triangle. Notice the location of the centroid. Is it in the interior or exterior of triangle or both??

You should have noticed that the centroid is always located in the interior of the triangle.

Now, construct the segment BG and GR and measure each along with segment BR.

Set up a ratio of the smaller segments to the median by calculating:

Notice the ratio. Drag point B to various locations. What do you notice about the ratios?

Can you make a conjecture about how the centroid divides the median? Explore further to see if this is true for the other two medians.

Write a summary about the medians of a triangle and the centroid of a triangle.

A GSP animation modeling all the concepts discussed in this assignment can be viewed after completion of the summary assignment.