1. **Problem statement:** Determine the perimeter of the right triangle with sides DE = 8.8 cm, hypotenuse DC = 10.6 cm, and angle ∠E = 34°. 2. **Formula and rules:** The perimeter of a triangle is the sum of the lengths of all its sides. 3. **Calculate the missing side EC:** Use the tangent function since $ \tan(34^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{EC}{DE} $. 4. **Calculate EC:** $$ EC = DE \times \tan(34^\circ) = 8.8 \times \tan(34^\circ) $$ Using a calculator, $ \tan(34^\circ) \approx 0.6745 $, so $$ EC = 8.8 \times 0.6745 = 5.94 \text{ cm (rounded to two decimals)} $$ 5. **Check the hypotenuse:** Given as 10.6 cm, verify with Pythagoras: $$ DE^2 + EC^2 = 8.8^2 + 5.94^2 = 77.44 + 35.28 = 112.72 $$ $$ \sqrt{112.72} = 10.61 \text{ cm} $$ This is very close to the given hypotenuse 10.6 cm, confirming the calculation. 6. **Calculate perimeter:** $$ P = DE + EC + DC = 8.8 + 5.94 + 10.6 = 25.34 \text{ cm} $$ Rounded to the nearest tenth: $$ P \approx 25.3 \text{ cm} $$ **Final answer:** The perimeter is approximately **25.3 cm**. Your calculation and conclusion are correct.