1. **Problem Statement:** Use the Pythagorean theorem to find the length of the hypotenuse $c$ in a right triangle with legs $a$ and $b$. 2. **Formula:** The Pythagorean theorem states that $$c^2 = a^2 + b^2$$ where $c$ is the hypotenuse and $a$, $b$ are the legs. 3. **Example:** Suppose $a=3$ and $b=4$. Substitute these values into the formula: $$c^2 = 3^2 + 4^2$$ 4. **Calculate squares:** $$c^2 = 9 + 16$$ 5. **Add:** $$c^2 = 25$$ 6. **Find $c$ by taking the square root:** $$c = \sqrt{25}$$ 7. **Simplify:** $$c = 5$$ **Answer:** The hypotenuse length is $5$. This test shows how to apply the Pythagorean theorem step-by-step to find the hypotenuse of a right triangle.