1. **Problem statement:** a. State the Pythagorean theorem correctly. b. Prove the Pythagorean theorem. 2. **Pythagorean theorem formula:** For a right triangle with legs $a$ and $b$ and hypotenuse $c$, the theorem states: $$c^2 = a^2 + b^2$$ 3. **Proof (using a geometric approach):** - Consider a right triangle with legs $a$ and $b$ and hypotenuse $c$. - Construct a square with side length $(a+b)$ and place four copies of the triangle inside it, arranged so that their hypotenuses form a smaller square in the center. - The area of the large square is: $$ (a+b)^2 = a^2 + 2ab + b^2 $$ - The four triangles each have area: $$ \frac{1}{2}ab $$ - The total area of the four triangles is: $$ 4 \times \frac{1}{2}ab = 2ab $$ - The smaller square formed by the hypotenuses has side length $c$, so its area is: $$ c^2 $$ - The area of the large square is also the sum of the areas of the four triangles plus the smaller square: $$ (a+b)^2 = 4 \times \frac{1}{2}ab + c^2 = 2ab + c^2 $$ - Equate the two expressions for the area: $$ a^2 + 2ab + b^2 = 2ab + c^2 $$ - Subtract $2ab$ from both sides: $$ a^2 + \cancel{2ab} + b^2 - \cancel{2ab} = c^2 $$ $$ a^2 + b^2 = c^2 $$ 4. **Explanation:** This shows that the square of the hypotenuse equals the sum of the squares of the other two sides, which is the Pythagorean theorem. **Final answer:** $$c^2 = a^2 + b^2$$