>> Welcome to a quick tutorial on the Pythagorean Theorem.
The Pythagorean Theorem states that in a right triangle the square of the hypotenuse is equal
to the sum of the squares of the other 2 sides, which are called "the legs."
So the hypotenuse is the longest side of a right triangle.
It's also opposite the right angle.
And it must be labeled "side C."
The reason that's so important is notice that C is on one side of the equation by itself.
The 2 legs are labeled A and B. And it doesn't matter which leg is A and which leg is B,
because notice they're on the same side of the equal sign.
So let's go ahead and call this "side A," and this "side B."
So the length of this side squared must equal the length of this side squared,
plus the length of this side squared.
I would like to spend a minute or 2 going over a proof for this theorem.
Now, there are several proofs for the Pythagorean Theorem.
I like the one that uses similar triangles, and here's how it works.
If you start with a larger right triangle, ABC, and you construct an altitude from vertex C
to side AB, you form 2 smaller right triangles that are similar to the larger right triangles.
Remember, in similar triangles the corresponding sides will be proportional.
So we're going to set up 2 proportions using these similar right triangles.
So first we're going to compare the ratio of A to C, which is the longer leg to the hypotenuse
of the largest right triangle, and that's equal to the ratio of E to A, which is the longer leg
of the blue right triangle to the hypotenuse of the blue right triangle.
And B is to C where B is the length of the shortest leg
of the largest right triangle, and C is the hypotenuse.
And that must equal the ratio of D to B, where D is the length of the smallest leg
of the smallest right triangle to its hypotenuse.
Remember, when you have a proportion, the cross products must be equal.
So we'd have A squared equals C times E, as we see here.
And we'd also have B squared equals C times D, as we see here.
So A squared plus B squared can be written as "C times E plus C times D."
Notice these 2 products have a common factor of C. So if we factor out C,
we're left with C times D plus E. But notice D plus E is just C. So we'd have C squared here,
and therefore the Pythagorean Theorem showing
that A squared plus B squared is equal to C squared.
Now that we've proven it, let's go ahead and try some problems.
We want to find the length of the missing side of the right triangle,
so we'll start with our theorem, C squared equals A squared plus B squared.
C must be the hypotenuse, which is this side here because it's opposite the right angle.
And these 2 would be the legs, so we'll label these "A" and "B."
Let's go ahead and substitute these values into the theorem, and then we can solve for C.
So we'd have C squared equals A squared or 12 squared, plus B squared or 9 squared.
So we have C squared equals 12 squared is 144.
Nine squared is equal to 81.
So we have C squared equals 225.
And now to find C we can take the square root of both sides.
Now, we're only going to take the principle square root or the positive square root here,
because we know C would be a length, and this length can't be negative.
Two hundred twenty-five is a perfect square, so we have C equals 15, in this case meters.
We have the length of the missing side.
Let's go ahead and try another one.
If you look at our diagram now we're trying to find the length of one of the legs.
So, again, we start with the Pythagorean Theorem.
This would be side C opposite the right angle.
Let's go ahead and call this "side A" and this "side B."
So using our Pythagorean Theorem we would have 22 squared must equal A squared,
which we don't know, plus 17 squared.
Twenty-two squared would be 484 must equal A squared plus 17 squared equals 289.
So now we have to isolate the A squared so we'll subtract 289 from both sides.
So that gives us 195 equals A squared.
And we'll square root both sides of the equation.
Again, we're only going to find the principle square root.
Now, 195 is not a perfect square so let's go to our calculator.
So second X squared brings up the square root, 195.
It's approximately 13.96, in this case centimeters.
So we found the length of the missing leg.
Let's go ahead and take a look at an application problem now.
Mason wants to lay pavers in his family's backyard for the summer.
It is important that he starts the pavers at a right angle.
If he wants the dimensions of the patio to be 8 feet by 10 feet,
what should the diagonal of the patio measure?
Okay; so we have a rectangle that's 8 feet by 10 feet.
Let's go ahead and sketch the diagonal.
Notice when we do this, we form 2 triangles.
And if we want this angle to be a right angle, then it must be true
that C squared must equal A squared plus B squared.
So using the Pythagorean Theorem if we find the length of C, we can determine the length
of the diagonal if this is a right angle.
So let's go ahead and try that.
C squared must equal 8 squared plus 10 squared if this is a right triangle,
and therefore forming a right angle for this patio.
So we'd have C squared equals 64 plus 100.
C squared equals 164.
And we can square root both sides of the equation.
C is going to be approximately the square root of 164.
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So it's approximately 12.81 feet.
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Now, since this will be measured using a tape measure, it might be helpful
to convert this .81 feet into inches.
Let's go ahead and do that on the side.
So .81 feet, put that over 1.
To convert this into inches, we'll multiply by 12 inches, which is equal to 1 foot.
Notice how the units of feet simplify out.
So we'll find this product that will give us the inches.
Point 81 times 12; so it's approximately 9.72 inches.
So C is approximately 12 feet 9.72 inches.
Point 72 is very close to .75, which would be 3/4 of an inch.
So Mason can get a very good indication what this length has to be with a tape measure
if he wants this to be a right angle.
I hope you found this video helpful.
Thank you for watching.
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