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Investigations with Centroids

 

By:Lauren Lee

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The centroid is the point of intersection of the medians of a triangle.

 

Medians are found by constructing segments from the vertices of the triangle to the midpoints of the sides opposite the vertices.So there will always be three medians in a triangle.

 

Letís look at one example of a median:

 

 

 

 

For our triangle ADC,B is the midpoint of the side AC.So DB is one median of the triangle.

 

 

 

Now letís look at the centroid of a triangle.

 

To do this, we will first have to construct the three medians of the triangle and then find the point of intersection.

 

 

 

 

In this example our medians are AE, BF, and CD.The point of intersection of the medians, which is our centroid, is labeled G because it is the center of gravity of the triangle.

 

 

 

Now that we know how to find the centroid, letís investigate!Will the centroid always be inside the triangle?

 

Letís look at three types of triangles:right triangle, acute triangle, and obtuse triangle.

 

 

 

Right Triangle

 

 

 

Acute Triangle

 

 

 

 

Obtuse Triangle

 

 

 

The centroid G is always inside the triangle, regardless the type, because it is the center of gravity.

 

 

 

 

Letís look at the medians of a triangle once more.

 

 

Hereís an example of a triangle with the distances between two points of a median noted below.

 

 

 

 

 

We can see that AG is twice as long as GE.Similarly, BG = 2GF and CG = 2GD.

 

 

So our final observation is that the medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.In other words, the longer segment of a median is always twice as long as the shorter segment.

 

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