## Exploration 4: Write-Up

GSP File

As shown in the picture, the centroid divides the triangle into 6 smaller triangles. As a result of the definition of median and its properties, the 6 smaller triangles actually have the same area.

As shown in the picture, the centroid divides the triangle into 6 smaller triangles. As a result of the definition of median and its properties, the 6 smaller triangles actually have the same area.

First, let’s explore a triangle split by one median. Let’s specifically look at the areas of other triangle.

Using the triangle above,  the point D is the midpoint of the segment AB. Thus, segment AD is congruent to segment DB. In addition, if segments AB and AD are considered the base of each triangle respectively, the height of each triangle will be the same. This is because the height is formed by taking a line perpendicular to the base and intersecting with the line parallel to the base passing through point C. Since both triangles have the same base and height, it follows that both triangles have the same area by definition of area of a triangle.

Now, let’s see if we can apply the 4 small triangles created by all three medians.

Looking at segment AC, it follows that segments AE and segments EC  are equal by definition of midpoint. Also, by the triangle midpoint segment theorem, it follows that segment DF is half of segment AC. Thus, segment DF is equivalent to segments AE and EC. Now, treating AE, EC, and DF as the bases of the triangles AED, ECF, DFB, and DEF respectively it follows that the height of each triangle is equivalent. This occurs because the segments AC and DF are parallel, and the line parallel to DF passing through point B are parallel. Because there is a unique distance between lines that are parallel and by the triangle midpoint segment theorem, the areas of each triangle are the same.