Final Exam

Part 3

by Debra Jackson

**3. Select one additional item from the assignments or from explorations presented in class that you have not written up. Submit a write-up about it.**

I chose problem 14 from assignment 4.

14. Prove that the three medians of a triangle are concurrent and that the point of concurrence, the centroid, is two-thirds the distance from each vertex to the opposite side.

How would you use GSP to help students understand this relationship of the triangle and its medians? How would you develop a sense of proof of the relationship with students?

1. Construct triangle ABC. | |
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2. Construct midpoints D and E of sides AB and AC. | |

3. Construct midsegment DE. | |

4. Construct segment BE and CD which intersect at point G. |

Because DE is the midsegment of △ABC, then DE = (1/2)BC. Because DE ∥ BC, then ∠EDC is congruent to ∠DCB and ∠DEB is congruent to ∠EBC since they are alternate interior angles. Therefore, △DEF ∼ △CBF by the AA similarity postulate. Because the △'s are similar, their sides are proportional making DF = (1/2)FC and EF = (1/2)FB. If DF = (1/2)FC, then DF = (1/3)DC and FC = (2/3)DC.

Similarly, this exercise can be done for the other two medians BE and AF.

Thus, it can be concluded that the medians of a triangle at concurrent at a point called a Centroid and that their lengths are divided into segments of length 1/3 and 2/3 with the longer length being between the vertex and the centroid.

When teaching the points of concurency in high school math courses, it can be difficult to help students understand the differences between the various types of points of concurrency as well as to understand the characteristics of each type. Going through this exercise will give the students another way of looking at and understanding these concepts as well as the ones used to prove the relationship of the parts of the medians.

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