 Let us look at a simple proof of the Pythagorean theorem that is that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the legs. Let us make this shape. We can do this by taking four copies of this triangle and rotating it 90 degrees counterclockwise to get the blue triangle and then the green one and then the orange one. The question is, is this nuke shape a square? First of all, the sides are equal which is good and we just have to check whether the interior angles are right. Well, since these are all right triangles, the sum of the other two must be 90 degrees and since this green angle here is congruent to this one, the sum of this orange angle must be equal to plus this one must be equal to 90 degrees. That applies to every corner of the square. Now, is that yellow shape inside a square? Well, all we have to check is well, we have to check two things that these angles are right and since these are all right triangles this angle must be right. And what about the sides? Yes, the sides are the same but what are those dimensions? Well, this length here is a this little piece here is b. So this side must be equal to a minus b. So we have a square of side a minus b. Hence, the area of the large square must be equal to the area of the four triangles plus the area of the yellow square. Well, the area of the large square is c squared. Well, the area of each triangle is a times b over two, we have four of those for so four times a b over two is equal to two a b. And the area of the yellow square, the side is a minus b. So it's a minus b squared. We do a quick computation here. This is equal to a square minus two a b plus b square. And these two terms collapse. And we get that c squared is equal to a square plus b square. Thank you.