Exploring Centroid of Triangles

By Princess Browne

 

        

     

       The centroid (G) of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.

Use Geometric Sketchpad (GSP) to construct the centroids and explore its location for various shapes of triangles.

 

First, I will construct the triangle and label the three vertices.   

      

 

Next, I will construct the midpoints for all 3 sides of the triangle and label each midpoint. On segments AB, AC, and BC are the midpoints D, E and F. The midpoint of AB is F, the midpoint of AC is D and the midpoint of BC is E. This implies that AF FB, AD DC and BE EC. The segment DF joins the midpoint of AB and AC. By the definition of a triangle mid-line DF is parallel to BC and DF is half of BC.

 

 

 

The medians are constructed by making segments from each vertex to the midpoint on the opposite side of the vertex. Finally, the intersection of all medians is constructed to form the centroid G.

 

 

 

 

 

lets take a closer look at the triangle.

 

AC = 5.54 cm

AB = 4.52 cm

BC = 6.16 cm

DF = 3.08 cm

DC = 5.41 cm

DG = 1.80 cm

GC = 3.60 cm

BF = 4.64 cm

BG = 3.09 cm

GF = 1.55 cm

AG = 2.67 cm

GE = 1.33 cm

 

DF/BC = FG/BG = 1/2

Show that FG = ½(BG)

 

 

When measuring the medians of a triangle we must consider the following formula. The distance from the midpoint of the opposite side of a vertex to the centroid and the distance from the vertex to the centroid have a ratio that is 1 to 2. The length of any vertex to the centroid (which is the larger piece of the medium) and the length of the entire medium will always have a ratio that is 2 to 3.

 

GC/DC = 2/3

 

 

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