If we remove the medians and construct a triangle from the medians, will
the new triangle be equilateral?
As we can see by the congruent side measures, the new triangle of medians
is also an equilateral triangle. Will this always be true? Moving the original
triangle so the measures of sides and medians change, we can record the
measures on a table. As can be seen, the chart indictes that the medians
of an equilateral triangle are equal and would therefore generate an equilateral
medians triangle. (This does not constitute a proof of deductive reasoning,
but is useful for students to see concretely the beginnings of such thought.)
If we our original triangle is isosceles, will the triangle of medians also
be an isosceles triangle?
As can be seen by the side measures, this triangle is isosceles. The
medians are represented in colors of red, green, and blue. If the medians
are removed and a new triangle is constructed, the new triangle of medians
is also isosceles.
Will the triangle of medians of and isosceles triangle always be isosceles?
We can use thefollowing table of measures and inductive reasoning that it
is likely so.
Will a right triangle of medians always generate a right triangle of
medians? The following triangle is a right triangle as can be see by the
measure of angle B.
When the medians are removed to form another triangle, it does not necessarily
follow that it will be a right triangle, as can be seen below.
By using parallel lines, we can construct the median triangle whose changes
coinside with the original triangles as it is manipulated. The following
construction illustrates both the original triangle and the median triangle
as right triangles.
At this point in the investigation, I would ask my students to manipulate
the triangle and use the tabulation to explore the ratios of the side measures
to determine under what conditions will the original triangle and median
triangle both be right triangle. Click here
to manipulate.