The Pythagorean Theorem is an important idea in geometry. It tells us that in a right triangle, which is a triangle with one angle that is 90 degrees, the square of the longest side (called the hypotenuse, or $c$) is the same as the sum of the squares of the other two sides (called $a$ and $b$). We can write this as:

$$c^2 = a^2 + b^2$$

How the Theorem Developed Over Time:

1. Babylonian Contributions (about 2000 BCE):

   • The earliest mentions of the theorem come from ancient Babylon. They had a tablet called Plimpton 322 that included a list of Pythagorean triples. This shows they understood how the sides of a right triangle relate to each other.

2. Ancient Egypt (about 1650 BCE):

   • An Egyptian document, known as the Rhind Mathematical Papyrus, shows a practical use of a 3-4-5 triangle. This triangle is a special example of a Pythagorean triple used for building things.

3. Indian Mathematics (about 800 CE):

   • An Indian mathematician named Baudhayana wrote about the theorem in his teachings, sharing rules that match the properties of right triangles.

4. Chinese Mathematics (about 300 CE):

   • An ancient Chinese book called the Zhou Bi Suan Jing talks about the Pythagorean Theorem, showing that people understood it well at that time.

5. Greek Mathematics:

   • The Greek mathematician Pythagoras (who lived from around 570 to 495 BCE) is often given credit for the first official proof of this theorem. However, it is likely that others already knew about it before him. His followers helped create a better understanding of geometry and came up with different proofs.

Ways to Prove the Theorem:

1. Geometric Proofs:

   • Visual Proof with Squares: One popular way to prove the theorem involves drawing squares on each side of the triangle. The area of the squares on the two shorter sides ($a$ and $b$) is the same as the area of the square on the longest side ($c$). This shows how the areas are connected.
   • Rearrangement Proof: This method shows that by changing the shape of the squares, we can demonstrate that $c^2$ equals the total of $a^2$ and $b^2$.

2. Algebraic Proofs:

   • Using algebra, we can represent the triangle's sides in the equation $c^2 = a^2 + b^2$ and manipulate the equation to prove that the theorem is true.

The Theorem’s Importance: The Pythagorean Theorem has played a big role in math and has been a crucial tool in many areas such as building design, science, and computer programming. Today, it is still very important in math classes, especially for ninth graders, helping them learn more about geometry.