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A Look at Medians

By

Tonya DeGeorge

 

In this investigation, we want to look at what happens when we construct a triangle made of the medians of an original triangle.  To begin, letÕs talk about what a median is.  The median of a triangle is the line segment connected from one of the vertices of a triangle to the midpoint of the opposite side of the vertex.  For example, consider the triangle below:

We can construct one of the medians by finding the midpoint of segment BC and connecting that to the vertex A, as shown below:

Similarly, we can find the other two medians:

However, what happens when we construct a triangle out of the medians of the original triangle?  And does it matter what type of triangle we work with?  To begin, letÕs look at an equilateral triangle.

Equilateral Triangle

First, we need to construct the equilateral triangle, which can easily be done in GeometerÕs Sketchpad.  Below are images of the equilateral triangle (left image) and the triangle constructed from the medians of the equilateral triangle, which is highlighted in yellow (right image):

                                                                                     

So what can we say about this new triangle?  Is this an equilateral triangle as well?  In order to answer this question, we first need to investigate the lengths of each of the three medians of the equilateral triangle.  To begin, letÕs look at the median (outlined in pink):

 

We already know that all the sides of this triangle are equal (all sides labeled as x), so how can we determine the length of the median?  Well, in this particular case, the median also happens to be the altitude of the triangle.  We can find the length of this altitude by using the Pythagorean Theorem.                                             

 

 

                                                                                                                                                                          

 

Similarly, we can find the lengths of the other two medians in the same manner.  Therefore, we know that all the medians must be the same length!  So if we construct a triangle from the medians of the equilateral triangle, then that triangle must be equilateral as well (by definition).

Isosceles Triangle

Now if we take a look at an isosceles triangle, we can see whether there is a relationship between the triangle created by the medians of the isosceles triangle and the isosceles triangle itself. Below are images of the isosceles triangle (left image) and the triangle constructed from the medians of the isosceles triangle, which is highlighted in yellow (right image):

                    

So what can we say about this new triangle?

Well, if we look at the relationships between the sides of the triangle, we can determine what type of triangle it is.  LetÕs first look at the original isosceles triangle:

                                                                                                                            

Knowing that this is an isosceles triangle, we know that AC = AB and .  However, we also know that BD=DA and AF=FC because D and F are the midpoints of AB and AC.  Therefore, we can say that  (by Side Angle Side) because they also share a common side (BC). 

                         

Therefore, we can also say that CD=BF due to CPCTC (Corresponding Parts of Congruent Triangles are Congruent).  Hence, we now know that the triangle that was constructed by the medians of the isosceles triangle is an isosceles triangle as well (also shown in the picture below).                                                                                                         

Right Triangles

Finally, letÕs look at right triangles and see if we get a right triangle that is constructed by the medians (from the last two examples, it looks like the original triangle gives a triangle just like it when we construct it using the medians – so letÕs see if this statement holds true for right triangles as well). Below are images of the right triangle  (left image) and the triangle constructed from the medians of the right triangle, which is highlighted in yellow (right image):

          

So what type of triangle does this form?  Well, we know that the triangle constructed by the medians is not an equilateral triangle (since none of the sides are equal) or an isosceles triangle (no two sides are equal).  Therefore, we can only conclude that the triangle formed by the medians is a scalene triangle (a triangle with no congruent sides).

However, there are also special cases to consider.  Suppose we have an isosceles right triangle (a right triangle where the two legs are congruent).  Knowing that the two legs are congruent and following the same method as we did when looking at an isosceles triangle, we know that the triangle that is constructed by the medians of the isosceles right triangle will also be an isosceles triangle (shown below). 

     

However, we can still see that the triangle formed is not a right triangle.  So when will the medians create a right triangle?

However, this does not mean that a right triangle will not produce another right triangle from its medians.  Through some investigation, we can see that a right triangle with angle measure will generate another right triangle formed by its medians (click here to move points and investigate angle measures).

In conclusion, we have found:

¯ All equilateral triangles form equilateral triangles from its medians

¯ All isosceles triangles form isosceles triangles from its medians

¯ Right triangles are special cases:

o   If the original triangle is an ordinary right triangle, the triangle formed is a scalene triangle

o   If the original triangle is a right triangle with angle measures , then the triangle formed is a right triangle as well with the same angle measures

o   If the original triangle is an isosceles right triangle, then the triangle formed is an isosceles triangle

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