Medians connect a vertex to the midpoint of the opposite side.
See solution.
Practice makes perfect
We will begin by drawing the following arbitrary triangle, â–ł ABC.
The median describes the point of concurrence of the lines from each vertex to the midpoint of the opposite side.
Finding the median
Let's use a ruler to find the midpoint of each side and then fold the paper along the vertex and its respective midpoint.
We will continue measuring the distance of another side of the triangle and fold it accordingly.
Finally, we will fold the third side in the same manner as before.
Once all of the medians have been folded, let's look at how all of the folds intersect.
Demonstrating the Theorem
Notice that all of the folds intersect at one point called the centroid. The Concurrency of Medians Theorem states that the medians of a triangle are concurrent at a point that is 23 the distance from each vertex to the midpoint of the opposite side. To investigate the theorem, let's measure the length from the vertex to the centroid for each median with our ruler.
As we can see, the distance of the median, AD, is 3.6 cm and the distance from the vertex to the centroid is 2.4 cm. let's also measure the distance of the rest of the medians and summarize everything below.
rrrr
AD:& 3.6 cm & AO: & 2.4 cm
BF:& 2.5 cm & OF: & 1.66 cm
CE:& 4.3 cm & OF: & 2.86 cm
Now we can test the Concurrency of Medians Theorem by dividing the length from each vertex to the centroid with the respective median.
rccc
AD:& 2.4 ? =2/3( 3.6) & ⇔ & 2.4=2.4 [0.8em]
BF:& 1.66? =2/3( 2.5) & ⇔ & 1.66=1.66 [0.8em]
CE:& 2.86? =2/3( 4.3) & ⇔ & 2.86=2.86
The theorem is correct.
