Today I am going to explore problem 2 on Jim Wilson's Assignment 6. During this exploration I plan to take a triangle, construct itís medians, and construct another triangle whose sides are the lengths of the medians of the first triangle. I plan to do this several times with different types of triangles and discover whether or not the two triangles have the same shape. For example, when the original triangle is an equilateral triangle, I want to know if the medians triangle will be an equilateral triangle. I also plan to explore an isosceles triangle and a right triangle. After I find an answer, I plan to figure out why that answer is true.
First I want to look at an equilateral triangle. From exploration I have discovered that both the original triangle and the medians triangle are equilateral.
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Also from exploration I have discovered that a triangle constructed so that the sides are the lengths of the medians of an isosceles triangle is also isosceles.
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Now from exploring a right triangle I have discovered that the medians triangle of the right triangle is not always a right triangle. I have discovered that the medians triangle is a right triangle only when two of the medians of the original triangle are perpendicular to each other.
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But why?
By comparing the angles formed by the medians of an equilateral triangle, an isosceles triangle, a right triangle, and a triangle free to take any kind of triangular shape with each of their medians triangles we see that each angle of the medians triangle is equal to two of the angles formed by the medians of each original triangle. So for the right triangleís medians triangle to be a right triangle then two of the medians must form 90 degree angles so that the angle of the medians triangle that corresponds to the 90 degree angles will be 90 degrees. Hence, the right triangle.
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But why are the angles formed from the medians of the original triangle equal to the angles of the medians triangle?
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From this proof we can justify all three situations. First, the medians triangle formed from the lengths of the medians of a equilateral triangle would also be equilateral because the medians of the original triangle would form 6 equal angles therefore the angles that correspond in the medians triangle would all be equal. Second, the medians triangle formed from the lengths of the medians of an isosceles triangle would also be isosceles because the medians of the original triangle would form at least 4 equal angles (6 only if it is also equilateral) so two of the angles of the medians triangle would be equal. Finally, as mentioned before, a triangle formed from the lengths of the medians of a right triangle will be right if and only if two of the medians of the original triangle are perpendicular.
But what kinds of right triangles have two medians that are perpendicular?
We can prove that the medians of a right triangle are perpendicular only if the angles of the right triangle are 90 degrees, 55 degrees, and 35 degrees.
Therefore, in conclusion, we have found that a triangle whose sides are constructed to be the same length of the medians of a right triangle is a right triangle if and only if the original triangle has angles that are 90 degrees, 55 degrees, and 35 degrees.