Visualizing the Pythagorean Theorem | Math with Mr. J

In this video I'm going to go through a visual representation of the Pythagorean Theorem in order to help us better understand the Pythagorean Theorem. Let's jump into our example where we have a right triangle with side lengths of 4 feet, 3 feet, and 5 feet. Remember we use the equation a squared plus b squared equals c squared when it comes to the Pythagorean Theorem. So let's start by finding 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. And remember, this relationship between sides is true for all right triangles. I hope that helped. Thanks so much for watching. Until next time, peace.