1. **Problem Statement:** Find the length of the hypotenuse $c$ in the right-angled triangle with legs 3 and 7, expressing the answer exactly using a surd. 2. **Formula:** Use the Pythagorean theorem for right-angled triangles: $$c^2 = a^2 + b^2$$ where $a$ and $b$ are the legs, and $c$ is the hypotenuse. 3. **Apply the formula:** Here, $a=3$ and $b=7$. $$c^2 = 3^2 + 7^2 = 9 + 49 = 58$$ 4. **Find $c$ by taking the square root:** $$c = \sqrt{58}$$ 5. **Simplify the surd if possible:** 58 factors as $2 \times 29$, both prime, so $\sqrt{58}$ is already in simplest surd form. 6. **Final answer:** $$c = \sqrt{58}$$ This means the hypotenuse length is exactly $\sqrt{58}$ units. --- **Explanation:** - The Pythagorean theorem relates the sides of a right triangle. - Square each leg, add them, then take the square root to find the hypotenuse. - Always check if the square root can be simplified by factoring out perfect squares. This method works for any right triangle when you know the two legs.