Alex Szatkowski
The centroid is one of the most commonly explored centers of a triangle. It is formed at the intersection of the medians of the triangle. A median can be constructed by joining the vertex of each angle to the opposite leg at its midpoint in a triangle.
First, lets prove that the medians are concurrent and intersect the medians two thirds the distance from from the vertex to the midpoint of the opposite side.
Let O be the origin, and point P be the point two thirds from the vertex A to midpoint M, point T be the point two thirds from the vertex C and midpoint R, and point Q be the point two thirds from the vertex C and midpoint N, using position vectors we can see that:
By using position vectors, I have proven that the points P,Q, and T are all the same point and divides the medians into two pieces with a 2 : 1 ratio.
Second, lets discuss the idea of the centroid being the center of mass.
The centroid is commonly referred to as the center of mass or gravity. Students can explore this with manipulatives, literally finding the point that the triangle can balance on a point. In addition, this point can be seen with other polygons.
Let us prove that the medians of a triangle partition the original triangle into 6 triangles with the same area.
Let us look at our triangle as being seperated into 6 different triangles.
Let P be the centroid, and AN = NC, PN = PN, and by definition of the centroid AP = PC
∴ △ ANP = △CNP by SSS congruency theorem.
Similarly, △ARP = △BRP and △BMP = △CMP
△ ACP = △ACM - △ CPM
△ ABP =△ABM - △ BPM
∴ △ ACP = △ABP and △ ANP = △NPC, △ ANP = △
ACP
Since △ARP = △BRP, △ ARP = △
∴ △ ARP = △BRP = △NCP = △ ANP and furthermore, △ARP = △BRP = △ NCP = △ANP = △CMP = △BMP
So point P, the centroid, is the center of mass for this triangle.
Extension: Using the understanding of the centroid being the center of mass for the triangle, we can look at the center of mass for a quadrilateral.
1. Divide the quadrilateral into two triangles using a diagonal.
2. Find the centroid for both triangles.
3. Repeat using the other diagonal, to find two more triangles and their centroids.
3. Once all found centroids have been found, connect the centroids with lines. Where the two lines intersect is the center of mass.
Finally, lets see how we can use the centroid to trisect a line segment.
1. Let AP be a line segment and a median of △ ABC. Let BR and CQ be the other medians.The point at which AP, BR, and CQ intersect is the centroid.
2.
Since the centroid is located two thirds the distance from the vertex to the midpoint of the opposite side,
line segment AB has been divided in a 2 : 1 manner. AO = 2OP.
3. In order to trisect AB, I now must find the midpoint of segment AO, thus partitioning AB into three equal line segments, or trisecting AB.
