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CCSS.Math: ,

So we have this rectangle
right over here, and we're told that the
length of AB is equal to 1. So that's labeled
right over there. AB is equal to 1. And then they tell us that
BE and BD trisect angle ABC. So BE and BD trisect angle ABC. So trisect means dividing
it into 3 equal angles. So that means that this
angle is equal to this angle is equal to that angle. And what they want
us to figure out is, what is the perimeter
of triangle BED? So it's kind of
this middle triangle in the rectangle
right over here. So at first this seems
like a pretty hard problem, because you're like well, what
is the width of this rectangle. How can I even start on this? They've only given
us one side here. But they've actually given
us a lot of information, given that we do know
this is a rectangle. We have four sides, and
that we have four angles. The sides are all
parallel to each other and that the angles
are all 90 degrees. Which is more than
enough information to know that this is
definitely a rectangle. And so one thing we do
know is that opposite sides of a rectangle are
the same length. So if this side is 1, then
this side right over there is also 1. The other thing we know is
that this angle is trisected. Now we know what the
measure of this angle is. It was a right angle, it
was a 90 degree angle. So if it's divided into three
equal parts, that tells us that this angle right
over here is 30 degrees, this angle right over
here is 30 degrees, and then this angle right
over here is 30 degrees. And then we see
that we're dealing with a couple of
30-60-90 triangles. This one is 30, 90, so this
other side right over here needs to be 60 degrees. This triangle right over here,
you have 30, you have 90, so this one has
to be 60 degrees. They have to add up to
180, 30-60-90 triangle. And you can also figure out
the measures of this triangle, although it's not going
to be a right triangle. But knowing what we know
about 30-60-90 triangles, if we just have
one side of them, we can actually figure
out the other sides. So for example, here we
have the shortest side. We have the side opposite
of the 30 degree side. Now, if the 30 degree side
is 1, then the 60 degree side is going to be square
root of 3 times that. So this length
right over here is going to be square root of 3. And that's pretty useful
because we now just figured out the length of the entire
base of this rectangle right over there. And we just used our knowledge
of 30-60-90 triangles. If that was a little
bit mysterious, how I came up with
that, I encourage you to watch that video. We know that 30-60-90
triangles, their sides are in the ratio of 1 to
square root of 3 to 2. So this is 1, this
is a 30 degree side, this is going to be square
root of 3 times that. And the hypotenuse
right over here is going to be 2 times that. So this length
right over here is going to be 2 times this
side right over here. So 2 times 1 is just 2. So that's pretty interesting. Let's see if we can
do something similar with this side right over here. Here the 1 is not the side
opposite the 30 degree side. Here the 1 is the side
opposite the 60 degree side. So once again, if we
multiply this side times square root of 3, we
should get this side right over here. This is the 60, remember this
1, this is the 60 degree side. So this has to be 1 square
root of 3 of this side. Let me write this down, 1
over the square root of 3. And the whole reason, the
way I was able to get this is, well, whatever this
side, if I multiply it by the square root of 3, I
should get this side right over here. I should get the 60
degree side, the side opposite the 60 degree angle. Or if I take the 60 degree
side, if I divide it by the square root of 3 I should
get the shortest side, the 30 degree side. So if I start with the
60 degree side, divide by the square root of 3, I
get that right over there. And then the
hypotenuse is always going to be twice the side
opposite the 30 degree angle. So this is the side opposite
the 30 degree angle. The hypotenuse is
always twice that. So this is the side opposite
the 30 degree angle. The hypotenuse is
going to be twice that. It is going to be 2 over
the square root of 3. So we're doing pretty good. We have to figure
out the perimeter of this inner triangle
right over here. We already figured
out one length is 2. We figured out another length
is 2 square roots of 3. And then all we have to really
figure out is, what ED is. And we can do that
because we know that AD is going to be
the same thing as BC. We know that this entire
length, because we're dealing with a rectangle,
is the square root of 3. If that entire length
is square root of 3, if this AE is 1 over
the square root of 3, then this length
right over here, ED is going to be
square root of 3 minus 1 over the square root of 3. That length minus that
length right over there. And how to find the perimeter
is pretty straight forward. We just have to add these
things up and simplify it. So it's going to be,
just let me write this, perimeter
of triangle BED is equal to-- This is
short for perimeter. I just didn't feel like
writing the whole word.-- is equal to 2 over the square
root of 3 plus square root of 3 minus 1 over the square
root of 3 plus 2. And now this just boils down
to simplifying radicals. You could take a
calculator out and get some type of decimal
approximation for it. Let's see, if we have 2 square
root of 3 minus 1 square root of 3, that will leave us with
1 over the square root of 3. 2 over the square of 3 minus
1 over the square root of 3 is 1 over the square root of 3. And then you have the
square root of 3 plus 2. And let's see, I can
rationalize this. If I multiply the numerator
and the denominator by the square root of 3,
this gives me the square root of 3 over 3 plus the
square root of 3, which I could rewrite
that as plus 3 square roots of 3 over 3. Right? I just multiplied this
times 3 over 3 plus 2. And so this gives us-- this
is the drum roll part now-- so one square root of 3
plus 3 square roots of 3, and all of that over 3, gives
us 4 square roots of 3 over 3 plus 2. Or you could put the 2 first. Some people like to write
the non-irrational part before the irrational part. But we're done. We figured out the perimeter. We figured out the perimeter
of this inner triangle BED, right there.