-
I promised you that I'd give
you some more Pythagorean
-
theorem problems, so I will
now give you more Pythagorean
-
theorem problems.
-
-
And once again, this is
all about practice.
-
Let's say I had a triangle--
that's an ugly looking right
-
triangle, let me draw another
one --and if I were to tell
-
you that that side is 7, the
side is 6, and I want to
-
figure out this side.
-
Well, we learned in the last
presentation: which of these
-
sides is the hypotenuse?
-
Well, here's the right angle,
so the side opposite the right
-
angle is the hypotenuse.
-
So what we want to do
is actually figure
-
out the hypotenuse.
-
So we know that 6 squared
plus 7 squared is equal to
-
the hypotenuse squared.
-
And in the Pythagorean theorem
they use C to represent the
-
hypotenuse, so we'll
use C here as well.
-
-
And 36 plus 49 is
equal to C squared.
-
-
85 is equal to C squared.
-
Or C is equal to the
square root of 85.
-
And this is the part that most
people have trouble with, is
-
actually simplifying
the radical.
-
So the square root of 85: can I
factor 85 so it's a product of
-
a perfect square and
another number?
-
85 isn't divisible by 4.
-
So it won't be divisible by 16
or any of the multiples of 4.
-
-
5 goes into 85 how many times?
-
No, that's not perfect
square, either.
-
I don't think 85 can be
factored further as a
-
product of a perfect
square and another number.
-
So you might correct
me; I might be wrong.
-
This might be good exercise for
you to do later, but as far as
-
I can tell we have
gotten our answer.
-
The answer here is the
square root of 85.
-
And if we actually wanted to
estimate what that is, let's
-
think about it: the square root
of 81 is 9, and the square root
-
of 100 is 10 , so it's some
place in between 9 and 10, and
-
it's probably a little
bit closer to 9.
-
So it's 9 point something,
something, something.
-
And that's a good reality
check; that makes sense.
-
If this side is 6, this side
is 7, 9 point something,
-
something, something makes
sense for that length.
-
Let me give you
another problem.
-
[DRAWING]
-
Let's say that this is 10 .
-
This is 3.
-
What is this side?
-
First, let's identify
our hypotenuse.
-
We have our right angle here,
so the side opposite the right
-
angle is the hypotenuse and
it's also the longest side.
-
So it's 10.
-
So 10 squared is equal to
the sum of the squares
-
of the other two sides.
-
This is equal to 3 squared--
let's call this A.
-
Pick it arbitrarily.
-
--plus A squared.
-
Well, this is 100, is equal to
9 plus A squared, or A squared
-
is equal to 100 minus 9.
-
A squared is equal to 91.
-
-
I don't think that can be
simplified further, either.
-
3 doesn't go into it.
-
I wonder, is 91 a prime number?
-
I'm not sure.
-
As far as I know, we're
done with this problem.
-
Let me give you another
problem, And actually, this
-
time I'm going to include one
extra step just to confuse you
-
because I think you're getting
this a little bit too easily.
-
Let's say I have a triangle.
-
-
And once again, we're dealing
all with right triangles now.
-
And never are you going to
attempt to use the Pythagorean
-
theorem unless you know for a
fact that's all right triangle.
-
-
But this example, we know
that this is right triangle.
-
If I would tell you the length
of this side is 5, and if our
-
tell you that this angle is 45
degrees, can we figure out the
-
other two sides of
this triangle?
-
Well, we can't use the
Pythagorean theorem directly
-
because the Pythagorean theorem
tells us that if have a right
-
triangle and we know two of the
sides that we can figure
-
out the third side.
-
Here we have a right
triangle and we only
-
know one of the sides.
-
So we can't figure out
the other two just yet.
-
But maybe we can use this extra
information right here, this 45
-
degrees, to figure out another
side, and then we'd be able
-
use the Pythagorean theorem.
-
Well, we know that the
angles in a triangle
-
add up to 180 degrees.
-
Well, hopefully you know
the angles in a triangle
-
add up to 180 degrees.
-
If you don't it's my fault
because I haven't taught
-
you that already.
-
So let's figure out what
the angles of this
-
triangle add up to.
-
Well, I mean we know they add
up to 180, but using that
-
information, we could figure
out what this angle is.
-
Because we know that this angle
is 90, this angle is 45.
-
So we say 45-- lets call this
angle x; I'm trying to make it
-
messy --45 plus 90--
this just symbolizes
-
a 90 degree angle --plus x
is equal to 180 degrees.
-
And that's because the
angles in a triangle always
-
add up to 180 degrees.
-
So if we just solve for x, we
get 135 plus x is equal to 180.
-
Subtract 135 from both sides.
-
We get x is equal
to 45 degrees.
-
Interesting.
-
x is also 45 degrees.
-
So we have a 90 degree angle
and two 45 degree angles.
-
Now I'm going to give you
another theorem that's not
-
named after the head
of a religion or the
-
founder of religion.
-
I actually don't think this
theorem doesn't have a name at.
-
All It's the fact that if I
have another triangle --I'm
-
going to draw another triangle
out here --where two of the
-
base angles are the same-- and
when I say base angle, I just
-
mean if these two angles are
the same, let's call it a.
-
They're both a --then the sides
that they don't share-- these
-
angles share this side, right?
-
--but if we look at the sides
that they don't share, we know
-
that these sides are equal.
-
I forgot what we call
this in geometry class.
-
Maybe I'll look it up in
another presentation;
-
I'll let you know.
-
But I got this far without
knowing what the name
-
of the theorem is.
-
And it makes sense; you don't
even need me to tell you that.
-
-
If I were to change one of
these angles, the length
-
would also change.
-
Or another way to think about
it, the only way-- no, I
-
don't confuse you too much.
-
But you can visually see that
if these two sides are the
-
same, then these two angles
are going to be the same.
-
If you changed one of these
sides' lengths, then the angles
-
will also change, or the angles
will not be equal anymore.
-
But I'll leave that for
you to think about.
-
But just take my word for it
right now that if two angles in
-
a triangle are equivalent, then
the sides that they don't share
-
are also equal in length.
-
Make sure you remember: not the
side that they share-- because
-
that can't be equal to anything
--it's the side that they don't
-
share are equal in length.
-
So here we have an example
where we have to equal angles.
-
They're both 45 degrees.
-
So that means that the sides
that they don't share-- this is
-
the side they share, right?
-
Both angle share this side --so
that means that the side that
-
they don't share are equal.
-
So this side is
equal to this side.
-
And I think you might be
experiencing an ah-hah
-
moment that right now.
-
Well this side is equal to this
side-- I gave you at the
-
beginning of this problem that
this side is equal to 5 --so
-
then we know that this
side is equal to 5.
-
And now we can do the
Pythagorean theorem.
-
We know this is the
hypotenuse, right?
-
-
So we can say 5 squared plus 5
squared is equal to-- let's say
-
C squared, where C is the
length of the hypotenuse --5
-
squared plus 5 squared-- that's
just 50 --is equal
-
to C squared.
-
And then we get C is equal
to the square root of 50.
-
And 50 is 2 times 25, so C is
equal to 5 square roots of 2.
-
Interesting.
-
So I think I might have given
you a lot of information there.
-
If you get confused, maybe you
want to re-watch this video.
-
But on the next video I'm
actually going to give you more
-
information about this type of
triangle, which is actually a
-
very common type of triangle
you'll see in geometry and
-
trigonometry 45,
45, 90 triangle.
-
And it makes sense why it's
called that because it has
-
45 degrees, 45 degrees,
and a 90 degree angle.
-
And I'll actually show you
a quick way of using that
-
information that it is a 45,
45, 90 degree triangle to
-
figure out the size if you're
given even one of the sides.
-
I hope I haven't confused you
too much, and I look forward
-
to seeing you in the
next presentation.
-
See you later.