1. The Centroid… and some properties Armando Martinez-Cruz Amartinez-cruz@fullerton.edu in consultation with Coach Karen Delaney Mathcoach@adelphia.net Buena Park Cluster June 25, 2005 TASEL-M

2. Mathematics and Pedagogy Proof Mathematical Reasoning Geometric Constructions Folding patty paper

3. Comments on learning difficulties (next four slides) Learners have difficulties recognizing triangles with the same area, especially when the triangles are completely different. The reasoning in this activity is based totally in this idea (triangles with the same area). Therefore, it is important that time is spent on this part before moving on.

4. In the next figure, let points C and D lie on a parallel to segment AB. Then, triangles ABC and ABD have the same area. Reason: They both have the same base (AB) and height (the distance from the parallel to the segment is unique).

6. Conversely, let A, B, C and D be four points on a line L. Let AB be congruent to CD. If E is a point not on the line L, the triangles ABE and CDE have the same area. Again, both triangles have the same base (since AB and CD are congruent segments) and the distance from a point to a line is unique.

8. The Medians in a Triangle A median in a triangle ABC is the segment that joints a vertex with the midpoint of the opposite side. Hence there are three medians in a triangle.

9. Medians are concurrent The medians in a triangle are concurrent (i.e., they meet in one interior point of the triangle.) The point of concurrency is the centroid of a triangle.

10. Geometric Constructions … folding (patty) paper At this point audience will use patty paper to determine the midpoint of a segment, the medians of a triangle, and their intersection point (the centroid).

11. The centroid has many properties: It is the center of gravity of the triangle It splits the triangle in six small triangles with the same area It splits the medians in the ratio 1:2

12. Center of Gravity The centroid is the center of gravity of the triangle (i.e., if the triangle is a sheet of iron and we want the triangle to be in balance over a sharp pin, we have to place the triangle, in such a way that the centroid is over the pin).

13. The area of all six little triangles is the SAME! The centroid splits the triangle in six small little triangles. All of them with the same area!!!

14. For instance, the blue and the yellow triangles have the same area since both bases are congruent (remember that Mc is a midpoint) and the distance from the centroid to AB is the –common--height for both triangles.

15. Notice that triangles CAMc and CMcB have the same area as well (again the base is the same and the height is the same).

16. So the orange triangle and the pink triangle have the same area (since we already proved that the yellow triangle and the blue triangle have the same area). And this in turn shows that all six little triangles have the same area.

17. The centroid splits each median into two segments, one being half the length of the other. From all previous work, triangle CAG is twice the area of triangle CGMa. And both have the same height! Let’s call it h. Let B be the base of CAG and b the base of CGMa. So the areas are related as 1/2Bh = 2(1/2)bh, which says that B = 2b as we wanted.