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hi i'm george woodbury from college of the sequoias in visalia california and in
this video i'm going to introduce you to the pythagorean theorem
one of the most widely known results in all of mathematics
the pythagorean theorem has to do with a right triangle
now a right triangle is a triangle that has a right angle as one of its three
angles
a right angle is a ninety degree angle
and you can think of it has
the perfect corner that you would see
in a square or a rectangle
if you think of it this way this looks a lot like a rectangle
that's been sliced in half diagonally
a little terminology -
the longest side of the triangle
opposite the right angle
is known as the hypotenuse of the triangle
the other two sides are simply called legs of the triangle
now if we label the two legs as a and b
in either order
and we labeled the hypotenuse as c
it needs to the result known as the pythagorean theorem
which relates
a, b, and c
in a formula
a squared plus b squared equals c squared
basically
if you take the length of the first leg and square it
and add that to the length
of the second leg
squared
that sum will equal the square of the hypotenuse
so again
that's a squared plus b squared equals c squared
typically
in a lot of problems we'll know the two legs
we'll square them, add them together, and then take the square root to figure out
what c is
or if we know the hypotenuse and one of the legs
we'll subtract the square of that leg over
and take the square root to find the other leg
let's take a look at some examples
two legs of a right triangle measure six centimeters and eight
centimeters
find the hypotenuse
let's begin by drawing a right triangle
a picture is always a great idea
and we're told
that the two legs are six and eight centimeters so
the unknown side is the hypotenuse
in the pythagorean theorem a squared plus b squared equals c squared
we can rewrite this as six squared
plus eight squared equals c squared
six squared is thirty six
eight squared is sixty four
and thirty six plus sixty four equals one hundred
now we have a squared variable
equal to a constant
so we can take the square root of both sides
now in previous situations we took both the positive and negative square
root
but in this geometry problem the hypotenuse cannot have negative length
so we're just going to worry about the positive square root and we find here
that ten is equal to c
slightly different problem here
one leg of a right triangle measure seven inches and the hypotenuse is
eleven inches long
find the length of the other leg
so we'll start with the right triangle again
not necessarily to scale, just so i can label the legs and the hypotenuse
one leg is seven inches let's put that here - i could have put it along on the bottom instead
the hypotenuse is eleven
and i'm missing this leg
i'll call it b
you could call it a if you like
again the pythagorean theorem - a squared plus b squared equals c squared
we'll fill in seven for a
and eleven for c
seven squared is forty nine
eleven squared - eleven times eleven - that's one hundred twenty one
subtract over the forty nine
b squared
equals
seventy two
so to solve for b
take the square root of both sides
again ignoring the negative square root because this has to be positive
b equals
i don't know the square root of seventy two
but i know that seventy two is the same as thirty six times two
so i can get a six out of there
six square root two is the exact length
and if i go to the calculator
that approximates to be
eight point fort eight five inches
by the way if you are just going to go ahead with a calculator approximation you could
have done it right here
at square root of seventy two which might be a little easier into the
calculator
uh... again if you know both legs you add that the squares together and then
take the square root
but if you know one leg and the hypotenuse like we did in this example
you'd have to do a little solving before you can take the square root
so if you have any questions or comments on this topic for this video or have a
request for a video that i can put together for you on youtube
you can reach me through the contact page on my website
and that address is george woodbury dot com (georgewoodbury.com)
thanks for watching and good luck with this