Explorations in Geometry
If you want to experiment and you don't have a copy of
the Geometer's sketchpad, you can click on this link to
try a Java site with an interactive graph started. Be warned
it may take a long time to load. Java site
Draw a triangle and find the midpoints of each side. From each vertex construct a segment to the midpoint of the opposite side. A segment like this is called the MEDIAN of a triangle. Find the point where these MEDIANS intersect.
1) Can you change the shape of the triangle so that the three medians do NOT intersect at the same point? Can you make them intersect outside the triangle.
2) For each of the medians, measure the distance from the intersection of the medians to the vertex, then measure the distance from the intersection to the side. Can you observe a pattern in these lengths that is true of ALL the medians?
3) Open a new sketch with a coordinate axis. Create a triangle with the three points having integer coordinates. Now find the coordinates of the intersection of the medians. Can you figure out a simple way to find the coordinates of the intersection from the coordinates of the vertices?
4) Explore the medians of triangles and see what else you can discover. Write a brief paragrah summarizing what you have found. Be sure to answer the specific questions above, and any other discoveries you have noted.
5) Here is an interesting property of medians you might have missed, draw a line through the point of intersection, Any line. Now measure the perpendicular distance to the three vertices of the triangle. This property has to do with one of the names commonly used for the intersection of the medians. Can you figure out which one?
6) Did you find a relationship between the length of the sides of the triangle and the length of the medians? Look closely and see if you can find this interesting constant relationship!!!