The explorations that follow deal with triangles and their triangle of medians. We start with a simple case, the equilateral triangle. As can be seen in Figure 1, the triangle of medians(the green shaded area) of the equilateral triangle is itself an equilateral triangle, since the sides of the triangle of medians are all the same length.
Now that you know what the triangle of medians looks like for the equilateral triangle, what would you expect the triangle of medians for an isosceles triangle to look like? Will the triangle of medians be an isosceles triangle? Figure 2 gives you the answer to these questions. As you can see from the lengths of the sides given below, AF=FG, thus the triangle of medians of an isosceles triangle is indeed an isosceles triangle. Is this what you expected?
Next, we will draw a right triangle. Would you expect the right triangle' s triangle of medians be a right triangle also? Figures 3 and 4 will give you some insite. In Figure 3, the right triangle's triangle of medians is a right triangle itself as can be seen by the angle measurements given in the figure. However, a right triangle does not always create a right triangle of medians. This is illustrated in Figure 4, angle ABC is 90 degrees but none of the angles of the triangle of medians is 90 degrees.
If the triangle of medians is a right triangle, would you expect the original triangle to be a right triangle also. Look at Figure's 5 and 6 below for illustrations to this question.
In Figure 5, the shaded region is the triangle of medians and is a right triangle. This can be seen by looking at the measure of angle ABC in the figure. Figure 5 also shows you that the original triangle(BDE) is a right triangle. As when the original triangle was a right triangle in Figure's 3 and 4, forming a right and non-right triangle of medians, respectively, the same is true when the triangle of medians is a right triangle. Figure 6 illustrates an instance when the triangle of medians is a right triangle but the original triangle(triangle BDE) is not a right triangle. This is noted in the measures of angles ABC and DBE in Figure 6.
Through these previous explorations, a question probably arose about when can one expect the two triangle to both be right triangles. It turns out that the only time when the original triangle and its triangle of medians are both right triangles is when the other two angles in both triangles are 35 degrees and 55 degrees. This point is shown in Figure 7.
If the reader would like to explore other similarities between the original triangle and its triangle of medians visit a GSP animation to help in the explorations.
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