1. **State the problem:** Calculate the surface area of a triangular prism with given dimensions: height of the triangle = 8 cm, base width = 6 cm, side width = 9 cm, and length of the prism = 10 cm. 2. **Formula for surface area of a triangular prism:** The surface area (SA) is the sum of the areas of the two triangular bases and the three rectangular faces. $$SA = 2 \times \text{Area of triangle} + \text{Perimeter of triangle} \times \text{Length}$$ 3. **Calculate the area of the triangular base:** The triangle has sides 6 cm, 8 cm, and 9 cm (height is 8 cm, base is 6 cm, side is 9 cm). We can use Heron's formula to find the area. First, find the semi-perimeter $s$: $$s = \frac{6 + 8 + 9}{2} = \frac{23}{2} = 11.5$$ Then, area $A$: $$A = \sqrt{s(s - 6)(s - 8)(s - 9)} = \sqrt{11.5(11.5 - 6)(11.5 - 8)(11.5 - 9)}$$ $$= \sqrt{11.5 \times 5.5 \times 3.5 \times 2.5} = \sqrt{11.5 \times 5.5 \times 3.5 \times 2.5}$$ Calculate inside the root: $$11.5 \times 5.5 = 63.25$$ $$3.5 \times 2.5 = 8.75$$ $$63.25 \times 8.75 = 553.4375$$ So, $$A = \sqrt{553.4375} \approx 23.53 \text{ cm}^2$$ 4. **Calculate the perimeter of the triangular base:** $$P = 6 + 8 + 9 = 23 \text{ cm}$$ 5. **Calculate the surface area:** $$SA = 2 \times 23.53 + 23 \times 10 = 47.06 + 230 = 277.06 \text{ cm}^2$$ **Final answer:** The surface area of the triangular prism is approximately **277.06 cm\textsuperscript{2}**.