## 6-1: Medians

Using some of the construction methods used in previous sections and definitions covered in this section we will explore Theorem 6-1.

Given the following triangle

Construct the centroid of the triangle then using the measurement tools show that Theorem 6-1 is accurate.

Step by Step Instruction

a. First we construct all the medians to locate the centroid. First we locate the midpoints of the sides of the triangles. (Highlight the sides of the triangles, from the "Construct" menu select "Midpoint". Then construct line segments from each vertex to the midpoint of the opposite line segment)

b. Once you have constructed the medians, locate the point where the three medians intersect this point is the centroid

c. Take time to explore that the centroid is actually concurrent no matter how you change the shape of the triangle.

Here we can see that no matter the shape of the triangle the medians always intersect at one point; therefore the centroid is always concurrent.

d. Now let's examine whether Theorem 6-1 is true.

Theorem 6-1 states that the length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint. We can interpret the theorem in two ways that the distance from the vertex to the centroid divided by the distance from the centroid to the midpoint is equal to 2 or the distance from the centroid to the midpoint divided by the distance from the vertex to the centroid is equal to .50. We can explore this by using the measure tools.

We will examine our original triangle and the three previous explorations.

Using the original triange, first we determine the distance from point E to the centroid. (Highlight point E and point C, from the "Measure" menu select "Distance". Then highlight point C and the midpoint of line segment DF, from the "Measure" menu select "Distance")

Then we calculate the quotient of measure of line segment EC divided by measure of line segment CH. (Highlight measures of each line segment, from the "Measure" menu select "Calculate". From the "Values" pull down menu select "m of line segment EC" then select the division key, then from the "Values" pull down menu again select "m of line segment CH") We see that the quotient is equal to 2.

Using the same method calculate the quotient of measure of line segment CH divided by measure of line segment EC. Here we see that the quotient is equal to .50.

Thus we have verified Theorem 6.1.

Do the same with the previous explored triangles to see if the Theorem is consistent with them as well

Yes, we have verified that Theorem 6-1 is true