3.8 The Centroid - Investigation 1: Are Medians Cocurrent?
Right Scalene Triangle - Medians
4.65/2.32 = 2/1
4.56/2.28 = 2/1
2.88/1.44 = 2/1
Acute Scalene Triangle - Medians
3.32/1.66 = 2/1
4.26/2.13 = 2/1
3.98/1.99 = 2/1
Obtuse Scalene Triangle - Medians
5.19/2.6 = 2/1
2.28/1.4 = 2/1
5.94/2.97 = 2/1
Median Concurrency Conjecture
The three medians of a triangle are concurrent.
Step 3 - Question
Is the centroid equidistant from the three vertices? From the three sides? Is the centroid the midpoint of each median?
Each vertices has a different distance from the centroid, the distance from the centroid and the sides are also different but they are half the distance from the vertices to the centroid. Although the centroid is not the midpoint of each median.
Step 5 - Question
Do you get the same ration for each median?
For all three of my triangles all my ratios when simplified were equal to 2/1. The numerator was always twice as large as the denominator.
Centroid Conjecture
The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the opposite side.