1. **State the problem:** We need to find the length of the hypotenuse $c$ of a right triangle with legs measuring 6 meters and 8 meters. 2. **Formula used:** According to the Pythagorean theorem, for a right triangle with legs $a$ and $b$ and hypotenuse $c$, the relationship is: $$c^2 = a^2 + b^2$$ 3. **Apply the formula:** Here, $a = 6$ m and $b = 8$ m. Substitute these values: $$c^2 = 6^2 + 8^2$$ 4. **Calculate squares:** $$c^2 = 36 + 64$$ $$c^2 = 100$$ 5. **Find $c$ by taking the square root:** $$c = \sqrt{100}$$ 6. **Simplify the square root:** $$c = 10$$ 7. **Answer:** The length of the hypotenuse is 10 meters. This is exact, so no rounding is necessary.