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Exploration #4

The Centroid and Area

By Annie Sun

This write up is to prove that the medians of a triangle divide the triangle into six smaller triangles with equal area.


Let's take a triangle ABC and construct the medians

The medians divide each side of the triangle into congruent segments, so that BD = CD, CE = AE, and AF = BF, marked on the triangle below

*The point of intersection of the three medians is called the centroid, labeled G

I'll start with showing small triangle BDG has equal area as the small triangle CDG

Let's take the two triangles and turn them to get a better look...

First, we know that the area of a triangle is 1/2*base*height; notice that both triangles BDG and CDG have the same base due to the fact that D is the midpoint of the line segment CB. Second, the altitude, or height, of both triangles is also the same when a perpendicular line segment is created from the "tops" of the triangles, at G, to the bases:

Therefore, because triangles BDG and CDG have the same base and height, then their areas (labeled x) are equal.

Now let's look at another set of small triangles, CEG and AEG, made from the medians of our original triangle ABC

Notice, like the previous two triangles that the smaller triangles have the same base, AE = CE, because E is the midpoint of AC. Also, the triangles share the same height, h, when the altitude is constructed from the point G and perpendicular to line segment AC.

Now because the two triangles have the same base and same height, their areas are equal and let's call their areas y.

Let's take the last set of smaller triangles, AFG and BFG.

Using the same logic as the first two sets of smaller triangles, we know the bases and the heights of triangles AFG and BFG are the same, thus their areas are equal; let's label the areas z.

Here is the original triangle with the sets of smaller triangles and their areas labeled accordingly.

To prove that the six smaller triangles have the same area, I now have to show that x = y = z.

Let's take a look at the 2 triangles AEB and CEB that make up the larger triangle ABC

Using the same concepts from the smaller triangles, we know triangles AEB and CEB have the same area because their bases are congruent, equal in value, and they have the same height, altitude.

Because the areas are equal then we know:

z + z + y = x + x + y

2z + y = 2x + y

2z = 2x

z = x or x = z

Now let's look at triangles ADB and ADC

Once again, the two triangles have the same base and height, thus because area of a triangle is 1/2*base*height, the two triangles have the same area


z + z + x = y + y + x

2z + x = 2y + x

2z = 2y

z = y or y = z

Using the transitive property, because it has been shown that x = z and z = y, then x = y or x = y = z.

Which means the smaller six triangles, made up of the medians of the larger triangle, all have the same area