Why is the centroid the balancing point of a triangle

Archimedes showed that the point where the medians are concurrent the centroid is the center of gravity of a triangular shape of uniform thickness and density. If you cut a triangle out of cardboard and balance it on a pointed object, such as a pencil, the pencil will mark the location of the triangle's centroid center of gravity or balance point. To locate the centroid through construction : We have seen how to construct a median of a triangle. Simply construct the three medians of the triangle.

The point where the medians intersect is the centroid. Actually, finding the intersection of only 2 medians will find the centroid.

Finding the third median, however, will ensure more accuracy of the find. Balancing a Triangle at the Centroid. So lets examine the areas of the two triangles created by each median. Lets look at the area of the six triangles created by the three medians, whose share the centroid G, as the common vertex of each triangle. One can notice, no matter the placement of A, B or C, the areas are always equal.

Thus, proving to me, that the centroid is the point of equilibrium, or a center of balance. The following explanation is how to construct a centroid for a pencil and paper construction.

If you know the coordinates of the triangle's vertices, you can calculate the coordinates of the centroid. See Coordinates of centroid. Home Contact About Subject Index. The point where the three medians of the triangle intersect. The circumcenter is equidistant from the vertices of the triangle. Once the inradius is known, each side of the triangle can be translated by the length of the inradius, and the intersection of the resulting three lines will be the incenter.

This, again, can be done using coordinate geometry. The midpoint is the point on the segment halfway between the endpoints.

It may be the case that the midpoint of a segment can be found simply by counting. If the segment is horizontal or vertical, you can find the midpoint by dividing the length of the segment by 2 and counting that value from either of the endpoints. To find out the coordinates of a point in the coordinate system you do the opposite.

Begin at the point and follow a vertical line either up or down to the x-axis. There is your x- coordinate. And then do the same but following a horizontal line to find the y- coordinate. This is the x-coordinate of the vertex.