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**Exploring Centroid of Triangles**

**By Princess Browne**

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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 Å FB, ADÅ
DC and BEÅ 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.

letÕs 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