1. **Problem Statement:** The Pythagorean theorem states that in a right triangle, the square of the hypotenuse length $c$ is equal to the sum of the squares of the other two sides $a$ and $b$. We use it to find the length of one side when the other two are known. 2. **Formula:** The theorem is expressed as $$a^2 + b^2 = c^2$$ where $a$ and $b$ are the legs and $c$ is the hypotenuse. 3. **Explanation:** This formula applies only to right triangles (one angle is 90 degrees). It helps us calculate distances and lengths in many real-world and geometric problems. 4. **Example:** If $a=3$ and $b=4$, then $$c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.$$ So the hypotenuse is 5 units long. 5. **Why we need it:** It allows us to find missing side lengths, calculate distances, and solve problems involving right triangles in fields like architecture, navigation, and physics. This theorem is fundamental in geometry and appears in many shapes and contexts beyond just simple triangles.