Median Centroid Right Triangle Example

So we're told that
AE is equal to 12.

That's this side
right over here.

And EC is equal to 18.

And then they've drawn a bunch
of the medians here for us.

So we know that they are medians
because, when they intersect

the opposite side, they're
telling us that this length is

equal to this length, or
that ED is equal to DC,

CB is equal to BA,
AF is equal to FE,

or that F, B, and
D are the midpoints

and that G, then, would
be the centroid where

the medians intersect.

And so the first thing they ask
us is, what is the area of BGC?

So BGC right here.

That is this triangle
right over there.

And to figure out
that area, we just

have to remind ourselves that
the three medians of a triangle

divide a triangle into six
triangles that have equal area.

So if we know the area of the
entire triangle-- and I think

we can figure this out.

This is a right triangle.

They're telling us that.

AE-- this entire distance right
over here-- is going to be 12.

So this is going to be 12.

Let me make sure I
have enough space.

This entire distance
right over here is 18.

They tell us that.

So the area of
AEC is going to be

equal to 1/2 times
the base-- which

is 18-- times the height--
which is 12-- which

is equal to 9 times
12, which is 108.

That's the area of this entire
right triangle, triangle AEC.

If we want the area of BGC or
any of these smaller of the six

triangles-- if we ignore
this little altitude

right over here, the ones that
are bounded by the medians--

then we just have
to divide this by 6.

Because they all
have equal area.

We've proven that
in a previous video.

So the area of BGC is
equal to the area of AEC,

the entire triangle, divided by
6, which is 108 divided by 6.

Which is what?

It's 60-- let's see.

You get 10 and then 48.

Looks like it would be 18.

It would be 18.

And that's right
because it would

be-- 108 is the same
thing as 18 times 6.

So we did our first part.

The area of that right
over there is 18.

And if we wanted,
we could say, hey,

the area of any of
these triangles--

the ones that are bounded
by the medians-- this

is going to be 18.

This is going to be 18.

This entire FGE triangle
is going to be 18,

but we did this first
part right over there.

Now they ask us, what
is the length of AG?

So AG is the distance.

It's the longer part of
this median right over here.

And to figure out
what AG is, we just

have to remind ourselves
that the centroid is always

2/3 along the way
of the medians,

or it divides the
median into two segments

that have a ratio of 2 to 1.

So if we know the entire
length of this median,

we could just take 2/3 of that.

And that'll give us
the length of AG.

And lucky for us, this
is a right triangle.

And we know that F and
D are the midpoints.

So for example, we
know this AE is 12.

That was given.

We know that ED is
half of this 18.

So ED right over here--
I'll do this in a new color.

ED is going to be 9.

So then we could just use
the Pythagorean theorem

to figure out what AD is.

AD is the hypotenuse
of this right triangle.

So we're looking at
triangle AED right now.

Let me write this down.

We know that 12
squared plus 9 squared

is going to be
equal to AD squared.

12 squared is 144.

144 plus 81.

And so this is going to
be equal to AD squared.

So this is what?

This is 225.

So we have 225 is
equal to AD squared.

And 225, you may or may not
recognize, is 15 squared.

So AD is equal to 15.

You want to take the principal
root, the positive root,

because we're talking about
distances or lengths of sides.

We don't care about
the negatives.

So AD is equal to 15.

So this whole thing
right over here

is going to be equal to 15.

And AG is going to be 2/3 of AD.

We proved that in
a previous video,

that the centroid is
2/3 along the way of any

of these medians.

And we could do it for
any of the medians.

So it's equal to 2/3 times
15, which is equal to 10.

So AG right over
here is equal to 10.

So we did the second part.

Now, this third part,
what is the area of FGH?

So let me color it in-- FGH.

So if we knew this length-- if
we knew HG and if we knew FH--

we could easily figure
out what that area is.

And there's actually
multiple ways

of figuring out either
one of those things.

So one way that we can think
about finding what HG is

is to remind
ourselves that HG is

the altitude of either
triangle FGE or triangle AFG.

And both of them
have a base of 6.

So this is 6, and
this is 6 over here.

And then they have a
height equal to GH.

And we know what the area is.

We know that the area
is already equal to 18.

So let's take this
triangle up here.

So we're talking about
the area of triangle AFG.

So we know it's 1/2
times its base, which

is 6, times its height,
which is GH-- times GH.

That's just 1/2
base times height

is equal to the area
of this triangle, which

is going to be equal to 18.

And so then, we just have
to tell ourselves, well,

this is-- 3 times
GH is equal to 18.

If we divide both sides of
this by 3, GH is equal to 6.

So that is one way to do it.

GH is equal to 6.

You could have also made
the similarity argument.

Then you could have said,
look, this triangle up here

is similar to this larger
triangle over here.

This hypotenuse is 2/3 of the
length of this entire thing.

So this is going to
be 2/3 of this 9.

So that's another way that
you could have gotten 6 there.

But either way, we
got this length.

Now we just have to
figure out what FH is.

And we could figure out what
FH is if we know what AH is.

Because we know A to F is 6.

So FH is going to
be AH minus AF.

So let's figure out what AH is.

Well, once again, we can
make a similarity argument.

And if we want to
do it formally,

we see that both this larger
right triangle and the smaller

right triangle, both have
a 90-degree angle there.

They both have this
angle in common.

So they have two
angles in common.

They are definitely
similar triangles.

And so we know the ratio of
AH-- let me do it in orange.

We know that the ratio
of AH to AE-- which

is 12-- is equal to
the ratio of AG-- which

is 10-- to the ratio of AD--
which we already figured out

was 15.

So one way to think about it
is AH is going to be 2/3 of 12.

Or we can just work
through the math just using

the similar triangles.

So this right-hand side
over here is just 2/3.

And so AH-- multiplying
both sides by 12--

is equal to 2/3 times
12, which is just 8.

So AH here is 8.

AH is 8.

AF is 6.

So FH right over here
is going to be 2.

And so now we have
enough information

to figure out the area of FHG.

So let me write it over here.

It's going to be
1/2 times the base.

I'll just use FH
as the base here,

although I could
do it either way.

Well, I'll use FH as the base.

1/2 times 2 times the height--
times 6-- which is equal to 6.

And we are done.

And you could keep going.

You could figure out the
length of pretty much

all of these segments here
using some of these techniques

or any of these areas.

Well, we've actually
figured out most of them.