1. **State the problem:** We have a right triangle formed by the TV tower, the guy wire, and the distance from the stake to the base of the tower. The tower height is 12 m, the guy wire (hypotenuse) is 14 m, and we need to find the horizontal distance from the stake to the tower base. 2. **Formula used:** In a right triangle, by the Pythagorean theorem, the relationship between the legs and the hypotenuse is: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse, and $a$, $b$ are the legs. 3. **Assign values:** Let $a = 12$ m (height of tower), $c = 14$ m (length of guy wire), and $b$ be the distance from the stake to the base (unknown). 4. **Apply the formula:** $$12^2 + b^2 = 14^2$$ $$144 + b^2 = 196$$ 5. **Solve for $b^2$:** $$b^2 = 196 - 144$$ $$b^2 = 52$$ 6. **Find $b$ by taking the square root:** $$b = \sqrt{52}$$ 7. **Simplify the square root:** $$b = \sqrt{4 \times 13} = \sqrt{4} \times \sqrt{13} = 2\sqrt{13}$$ 8. **Calculate the decimal value:** $$b \approx 2 \times 3.605551275 = 7.21110255$$ 9. **Round to one decimal place:** $$b \approx 7.2$$ **Final answer:** The stake is approximately 7.2 meters from the base of the TV tower.