1. **State the problem:** We need to find the perimeter of a right triangle with a vertical height of 9 cm and a base divided into two segments of 4 cm and 12 cm. 2. **Identify the sides:** The base of the triangle is the sum of the two segments: $$4 + 12 = 16\text{ cm}$$. 3. **Use the Pythagorean theorem:** Since the triangle has a right angle, the hypotenuse (the third side) can be found using $$a^2 + b^2 = c^2$$, where $$a = 9$$ cm (height), $$b = 16$$ cm (base), and $$c$$ is the hypotenuse. 4. **Calculate the hypotenuse:** $$c = \sqrt{9^2 + 16^2} = \sqrt{81 + 256} = \sqrt{337} \approx 18.3576\text{ cm}$$ 5. **Calculate the perimeter:** The perimeter $$P$$ is the sum of all three sides: $$P = 9 + 16 + 18.3576 = 43.3576\text{ cm}$$ 6. **Round the answer:** To 1 decimal place, the perimeter is: $$P \approx 43.4\text{ cm}$$ **Final answer:** The perimeter of the triangle is **43.4 cm**.