1. **State the problem:** We need to find the length of the hypotenuse $F$ of a right triangle with legs measuring 34 and 36 units. 2. **Formula used:** According to the Pythagorean theorem, for a right triangle with legs $a$ and $b$ and hypotenuse $c$, the relationship is: $$c = \sqrt{a^2 + b^2}$$ 3. **Apply the formula:** Here, $a = 34$ and $b = 36$, so: $$F = \sqrt{34^2 + 36^2}$$ 4. **Calculate the squares:** $$34^2 = 1156$$ $$36^2 = 1296$$ 5. **Sum the squares:** $$1156 + 1296 = 2452$$ 6. **Find the square root:** $$F = \sqrt{2452}$$ 7. **Simplify the square root:** $$2452 = 4 \times 613$$ $$F = \sqrt{4 \times 613} = \sqrt{4} \times \sqrt{613} = 2\sqrt{613}$$ 8. **Approximate the value:** $$\sqrt{613} \approx 24.7588$$ $$F \approx 2 \times 24.7588 = 49.5176$$ **Final answer:** $$F \approx 49.52$$