Welcome to Math with Mr. J. In this video, I'm going to go through an introduction to the Pythagorean Theorem. Now, the Pythagorean Theorem has to do with right triangles and the relationship between the sides of right triangles. It's called the Pythagorean Theorem because it's named after Pythagoras, a Greek philosopher and mathematician.
Let's jump into our examples and see exactly what this all means and looks like, starting with number one, where we have a right triangle. Now remember, the Pythagorean theorem applies to only right triangles. Before we get started with the specifics of the Pythagorean theorem, we need to take a look at the sides of this triangle, and we are going to start with this side right here. The side.
directly across from the right angle. This is called the hypotenuse. The hypotenuse is the longest side of a right triangle, and again it will be across from or opposite of the right angle.
This is something we need to recognize and know when it comes to the Pythagorean theorem. Then we have the other two shorter sides. So this side right here and this side right here. These are called the legs.
So this is a leg and this is a leg. The Pythagorean theorem states that the sum of the legs squared will equal the hypotenuse squared. So the lengths of the legs squared, add those together and that will equal the hypotenuse squared.
And that probably sounds confusing worded like that. So let's write it out as an equation. a squared plus b squared equals c squared.
So for the Pythagorean theorem, we use that equation. Again, a squared plus b squared equals c squared. Now a, b, and c all represent a side of the triangle.
Let's start with c. Now c is always going to be the hypotenuse. So let's put a c here. And then a and b are going to be the legs.
It does not matter which leg is a and which is b. It will work out the same either way. So let's call this a and this b. So what we are going to do, we are going to use the Pythagorean theorem, the equation a squared plus b squared equals c squared. to figure out the missing side length.
This side right here, the hypotenuse. If we know two of the side lengths, we can use the Pythagorean theorem to figure out the missing side length. Let's plug in the information we know in order to figure out the information we don't know.
So we have both of the legs given a and b. So let's plug those into the equation. So a squared plus b squared equals c squared. Again we are given a and b so let's plug those in.
A is four feet so four feet squared plus b is three feet so three feet squared equals c squared. Now we can work through this equation and solve for c so we need to figure out what c equals. Let's start with the left side of the equation. So 4 squared plus 3 squared. 4 squared means 4 times 4, so that gives us 16. Plus 3 squared, that means 3 times 3, that gives us 9, equals c squared.
16 plus 9, that equals 25, equals c squared. Now we need to isolate. that variable of c and get rid of the exponent of 2. We do that by taking the square root. So let's take the square root of c squared. Now whatever we do to one side of the equation, we must do to the other.
So let's take the square root of 25 as well. Now as far as the right side of the equation, the variable of c is now isolated. And then for the left side of the equation, the square root of 25 is 5. So c equals 5. Let's rewrite that with the variable first.
So c equals 5 and this is feet. So that is our missing side length. This is 5 feet right here.
We used the Pythagorean theorem to figure out the missing side length of that triangle. Now let's take a look at a visual representation of number one and the Pythagorean theorem. This is going to help us better understand the Pythagorean theorem.
For number one, we had a right triangle with legs that measured four feet and three feet. The hypotenuse measured five feet. So here is that right triangle. Let's find A, B, and C.
We will start with the legs. This is A right here, and this is B right here. Remember, A and B are always going to be the legs, and it doesn't matter which leg is A and which leg is B.
They are interchangeable, so keep that in mind. And then we have the hypotenuse, which is always C. The hypotenuse is the longest side, the side across from or opposite of the right angle.
So this is C. Now let's take all of those sides of this triangle and square them. And we're actually going to make a square on each side.
This is A right here. So A. This is B. So B.
and then this is C right here. So C. The areas of the two smaller squares, the legs, actually add up to the area of the large square, the hypotenuse. So the two smaller squares combined equal the large square. So the sum of the legs squared, so square those side lengths and add them together, and that sum is going to equal the hypotenuse squared, so that side length squared.
That's what the Pythagorean theorem states. So let's square each side length to find the area of each square on the sides of the triangle to show that this is true. For A, the area of that square is 16 square feet. For B, the area of that square is 9 square feet.
And then for C, the area of that square is 25 square feet. So again, the areas of the two smaller squares, the legs, add up to the area of the large square, the hypotenuse. 16 square feet plus 9 square feet equals 25 square feet.
So A squared plus B squared equals C squared. So it's pretty cool how that relationship works out for every right triangle. Now let's plug in a, b, and c into the equation to write it out that way as well. So we have a squared plus b squared equals c squared. Now we can plug in a, b, and c.
So a is 4 feet, so 4 squared. b is 3 feet, so 3 squared. plus c is 5 feet, so 5 squared. 4 squared is 16, plus 3 squared is 9, plus 5 squared is 25. 16 plus 9 is 25, so 25 equals 25. Now obviously that's true, 25 does equal 25, so the relationship between the sides holds true.
through that equation. We have the legs represented on the left side of the equation, a squared plus b squared. The sum of those legs squared was 25. And then the hypotenuse is represented on the right side of the equation.
We have c squared. The hypotenuse squared was also 25. So there you have it. There is a visual representation of the Pythagorean theorem.
Now let's move on to number two. For number two, we have a right triangle with given side lengths of 15 centimeters and 17 centimeters, and then we have a missing side length. Now for this one, we have a leg given and the hypotenuse given. So let's call this A, this B, so this is the missing side length, and then this C. Remember, C always has to be the hypotenuse.
And then A and B are the legs. It doesn't matter which leg is A and which is B. Now we can plug in what we are given into the equation a squared plus b squared equals c squared and solve for the missing side length.
So a squared plus b squared equals c squared. While we are given a 15 centimeters, so 15 centimeters squared plus b squared, we need to figure out what B is so leave it as B squared equals C squared while C is 17 centimeters so 17 centimeters squared now let's work through this equation and figure out what B equals we will start with 15 squared that means 15 times 15 that gives us 225 plus B squared equals 17 squared. That means 17 times 17. That gives us 289. Now we need to continue to work to isolate that variable.
So let's subtract 225 from the left side of the equation. Whatever we do to one side of the equation, we must do to the other. So let's subtract 225 from this side of the equation as well. The 225s on the left side of the equation cancel each other out, so we have b squared equals, and then on the right side of the equation, we have 289 minus 225. That equals 64. So we have b squared equals 64. We need to isolate that variable of b.
Since we are squaring b, we have an exponent of b squared. 2 we need to take the square root in order to isolate that B. Whatever we do to one side of the equation we must do to the other so we have the square root of 64 as well.
The B is now isolated equals and then the square root of 64 is 8 so B equals 8 and this is centimeters. This is our missing side length. So b is 8 centimeters. So there you have it. There's an introduction to the Pythagorean theorem.
I hope that helped. Thanks so much for watching. Until next time, peace.