Welcome to a lesson on the medians of a triangle.
The goals of this video are to define the medians of a triangle.
And then state the properties of the medians of a triangle.
The median is a line segment that joins a vertex.
And the midpoint of the opposite side.
So here we see a median in red.
And since this is the midpoint of the opposite side we know that this segment is congruent to this segment.
because each triangle has three vertices.
every triangle also has three medians.
And the point of concurrency
of the three median is called the centroid.
So in this diagram
this point here is the centroid.
It's the point where all three medians intersect.
The centroid is also the center of mass of an object of uniform density
in the shape of a triangle.
Which means the object would balance on any line through the centroid.
The way I like to think about this is
if this triangle was a piece of sheet metal with uniform density
then you place it horizontally.
Let's say this was the centroid.
If you put the tip of a rod
on the centroid underneath
this piece of sheet metal
it will balance perfectly horizontally
on that rod.
Now let's take a look at the concurrency of medians theorem.
The medians intersect in a point
that is two thirds the distance from the vertices
to the midpoint of the opposite side.
And we now know that point is the centroid.
So if we take a look at the median CD.
If we know the length is 27 units.
CM would be two third the total distance.
Well two third of 27 would be 18.
Leaving one through the distance for the segment
MD
that would be nine.
So this is another property that this theorem tells us.
So if we know that segment AE has a length of 15 units.
Then AM would be two third of fifteen that would be ten.
Leaving one third the total length
for ME so that be five.
And if we know the median BF is 12 units.
Here in green.
With two third of 12 would be eight.
So this would be eight units.
Two third of 12.
And this would be one third of 12 which would give us four.
Let's go and state this more formally.
AM would be two third of AE.
And ME
would be one third of AE.
There's also one more connection that we can make.
If we compare AM
and ME notice that ME is
half the length of AM.
And this property will also always hold true.
Hopefully that make sense if AM is two third AE.
and ME is one third AE.
ME is always gonna be half of AM.
And we can say the same properties for each median.
If we take a look at the median CD.
Here in orange.
CM is going to be two third CD.
And MD will be one third CD.
And again it follows that
MD will be one half of CM.
Let's go and take a look at the last median.
In green
BF.
BM will be two third BF.
So MF
would be equal to one third
BF.
And then finally
MF will be one half of BM.
I think we'll go ahead and stop here on this video.
I hope you found this helpful.