The Pythagorean theorem is a fundamental concept in mathematics that relates to the relationships between the sides of a right triangle. It is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery.

The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the adjacent and opposite sides). Mathematically, this can be expressed as:

c² = a² + b²

where c is the length of the hypotenuse and a and b are the lengths of the other two sides.

To understand the logic behind the Pythagorean theorem, it is helpful to consider a right triangle in a two-dimensional coordinate system. Let's label the vertices of the triangle as A, B, and C, where C is the vertex that is opposite the right angle.

Next, let's draw a square with sides of length c that is centered at point C. This square will have four corners that lie on the sides of the triangle.

We can then draw squares with sides of length a and b on the other two sides of the triangle. These squares will also have corners that lie on the sides of the triangle.

Now, we can see that the square with side length c is made up of four congruent right triangles, each of which has sides of length a and b. Similarly, the squares with side lengths a and b are each made up of four congruent right triangles.

Therefore, the total area of the square with side length c is equal to the sum of the areas of the squares with side lengths a and b:

c² = a² + b²

This is the Pythagorean theorem.

Another way to understand the Pythagorean theorem is to consider the distance formula in a two-dimensional coordinate system. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:

distance = √((x2 - x1)² + (y2 - y1)²)

If we have a right triangle with vertices at (0,0), (a,0), and (0,b), then the distance between the first two vertices is a and the distance between the second two vertices is b. The distance between the first and third vertices, which is the length of the hypotenuse, is given by:

c = √(a² + b²)

This again leads to the Pythagorean theorem.

Overall, the Pythagorean theorem is a fundamental concept in mathematics that relates to the relationships between the sides of a right triangle. Its logic can be understood through visualization and the distance formula in a two-dimensional coordinate system.
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