The Pythagorean Theorem states that in a right-angled 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. This can be expressed as:

\[ c^2 = a^2 + b^2 \]

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

There are several ways to prove the Pythagorean Theorem. Here, I'll show you a classic geometric proof using rearrangement.

### Proof by Rearrangement

1. **Construct two identical squares:**

   - Start with a square with side length \((a + b)\). Thus, the area of this square is \((a + b)^2\).
   - Inside this square, place four identical right-angled triangles with sides \(a\), \(b\), and hypotenuse \(c\).

2. **Calculate the area in two ways:**

   - The total area of the large square is \((a + b)^2\).
   - The area can also be calculated by adding the area of the four triangles and the area of the smaller square (with side \(c\)) in the center.

3. **Express the areas:**

   - The area of the large square is \((a + b)^2\).
   - The area of one triangle is \(\frac{1}{2}ab\). Therefore, the area of four triangles is \(4 \times \frac{1}{2}ab = 2ab\).
   - The area of the smaller square in the center is \(c^2\).

4. **Set up the equation:**

   Since both expressions represent the area of the large square, we can set them equal to each other:

   \[
   (a + b)^2 = 4 \times \left(\frac{1}{2}ab\right) + c^2
   \]

   Simplify the right-hand side:

   \[
   (a + b)^2 = 2ab + c^2
   \]

5. **Expand and simplify:**

   Expand the left-hand side:

   \[
   a^2 + 2ab + b^2 = 2ab + c^2
   \]

   Subtract \(2ab\) from both sides:

   \[
   a^2 + b^2 = c^2
   \]

This completes the proof of the Pythagorean Theorem.

Would you like more details or have any questions about this proof? Here are five questions you might consider asking next:

1. What are some other proofs of the Pythagorean Theorem?
2. How can the Pythagorean Theorem be applied in real-world problems?
3. Can the Pythagorean Theorem be extended to three dimensions?
4. What is the history behind the Pythagorean Theorem?
5. How can the Pythagorean Theorem be used in coordinate geometry?

**Tip:** When proving mathematical theorems, always check your work by considering different approaches to ensure the accuracy and robustness of your proof.

Discover how to prove the Pythagorean Theorem in geometry with a detailed step-by-step explanation. This theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Suitable for Grades 7-9.

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