Interactive Geometry Dictionary
Properties and Applications of the MedianWhat can you say about the three
medians of a triangle?
In the applet below, the three medians of a triangle have
been constructed. Drag the vertices to change the triangle. What is always
true of the three medians of a triangle?
Theorem
The three medians of a triangle all pass through one point. This point
is called the centroid of the triangle.
If the triangle were cut out of material with uniform density,
the centroid would be its center of gravity.
What is true about the sum of the lengths of
the three medians of a triangle?
A triangle and its three medians are given in the applet given below.
- The top segment has the same length as the perimeter of the triangle.
- The middle segment is the same length, but it has markers showing
1/4, 1/2, and 3/4 of its length.
- The third segment has the same length as the sum of the lengths of
the three medians of the triangle.
Drag any of the vertices to change the shape of the triangle and see
how the bottom segment varies with respect to the segment representing
the perimeter of the triangle. It is more important to change the shape
of the triangle than its size.
- How large can you get the bottom line to be with respect to the perimeter?
- How small can you get it to be?
- What conclusion can you make about how the sum of the lengths of the
three medians is related to the perimeter of the triangle?
Theorem
The sum of the
lengths of the three medians of a triangle is more than 3/4 p and
less than p, where p is the perimeter of the triangle.
The exact values of 3/4 p and p are attained
only for a degenerate triangle, when all three vertices are collinear.
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Pre-K - 2 
