1. **State the problem:** We need to find the perimeter of a right triangle with a height of 9 cm perpendicular to the base, which is 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: $4 + 12 = 16$ cm. 3. **Find the hypotenuse:** The hypotenuse is the side opposite the right angle. We can find it using the Pythagorean theorem: $$\text{hypotenuse} = \sqrt{(\text{height})^2 + (\text{base})^2} = \sqrt{9^2 + 16^2}$$ 4. **Calculate the hypotenuse:** $$\sqrt{81 + 256} = \sqrt{337}$$ 5. **Approximate the hypotenuse:** $$\sqrt{337} \approx 18.3576$$ 6. **Calculate the perimeter:** The perimeter is the sum of all three sides: $$9 + 16 + 18.3576 = 43.3576$$ 7. **Round to 1 decimal place:** $$43.4$$ **Final answer:** The perimeter of the triangle is $43.4$ cm.