To prove that $a^2 + b^2 = c^2$, we can analyze the square diagram shown in the image.

Explanation:

1. Understanding the Diagram:

   • The large square has side length $a + b$ (since each side is made up of one segment of length $a$ and another of length $b$).
   • Inside this large square, there is a rotated square formed by joining the vertices of the right triangles, with each side of this inner square being the hypotenuse $c$ of the right triangles.

2. Area Calculation:

   • Area of the Large Square: The area of the large square can be calculated by squaring its side length: $\text{Area of large square} = (a + b)^2$
   • Area in Terms of Triangles and Inner Square: The area of the large square can also be expressed as the sum of the areas of the four right triangles and the area of the inner square: $\text{Area of large square} = 4 \times \frac{1}{2} \times a \times b + c^2 = 2ab + c^2$

3. Setting Up the Equation:

   • Equating the two expressions for the area of the large square, we get: $(a + b)^2 = 2ab + c^2$

4. Expanding and Simplifying:

   • Expanding $(a + b)^2$: $a^2 + 2ab + b^2 = 2ab + c^2$
   • Subtract $2ab$ from both sides: $a^2 + b^2 = c^2$

This proves that $a^2 + b^2 = c^2$, which is the Pythagorean theorem.

Answer for the Question:

The highlighted segments in red represent the sides of the large square, which is formed by the legs $a$ and $b$ of the right triangles. These segments help establish that the side of the large square is $a + b$, which is essential for setting up the area-based proof.

Would you like more details, or do you have any questions about this proof?

Here are five questions to extend your understanding:

1. How does this proof relate to the Pythagorean theorem in other geometric shapes?
2. What are other methods for proving the Pythagorean theorem?
3. Why is it important to use area calculations in this proof?
4. How would the proof change if one of the legs was longer than the hypotenuse?
5. What real-world applications utilize the Pythagorean theorem?

Tip: When proving geometric theorems, breaking down the figure into simpler components (like triangles and squares) can often simplify the calculation process.