1. **State the problem:** We need to find the perimeter of a right-angled triangle with one leg measuring 3 m and the hypotenuse measuring 14 m. The other leg length is unknown. 2. **Recall the Pythagorean theorem:** For a right-angled triangle with legs $a$ and $b$, and hypotenuse $c$, the relationship is: $$a^2 + b^2 = c^2$$ 3. **Assign known values:** Let the unknown leg be $b$, the known leg $a = 3$, and the hypotenuse $c = 14$. 4. **Apply the Pythagorean theorem:** $$3^2 + b^2 = 14^2$$ $$9 + b^2 = 196$$ 5. **Solve for $b^2$:** $$b^2 = 196 - 9$$ $$b^2 = 187$$ 6. **Find $b$ by taking the square root:** $$b = \sqrt{187}$$ 7. **Calculate the approximate value:** $$b \approx 13.6748$$ 8. **Calculate the perimeter $P$:** $$P = a + b + c = 3 + 13.6748 + 14$$ $$P = 30.6748$$ 9. **Round to 1 decimal place:** $$P \approx 30.7$$ **Final answer:** The perimeter of the triangle is approximately **30.7 m**.