To find the surface area of the triangular prism, we need to calculate the area of the triangular face and the area of the rectangular faces.
Step 1: Calculate the area of the triangular face
The triangular face of the prism is a right triangle. The area \( A \) of a right triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] In this triangle:
- The base is \( 6 \) cm
- The height is \( 8 \) cm
Substituting in these values: \[ A = \frac{1}{2} \times 6 \times 8 = \frac{1}{2} \times 48 = 24 \text{ square centimeters} \]
Step 2: Calculate the area of the rectangular faces
The prism has three rectangular faces:
- The rectangular face on the right (height = 9 cm, width = length of the triangle's base = 6 cm).
- The rectangular face on the left (height = 9 cm, width = triangle's left side = 8 cm).
- The rectangular face forming the hypotenuse of the triangle (height = 9 cm, width = triangle's hypotenuse = 10 cm).
Area of the right rectangular face: \[ A_1 = \text{base} \times \text{height} = 6 \times 9 = 54 \text{ square centimeters} \]
Area of the left rectangular face: \[ A_2 = 8 \times 9 = 72 \text{ square centimeters} \]
Area of the hypotenuse rectangular face: \[ A_3 = 10 \times 9 = 90 \text{ square centimeters} \]
Step 3: Total area of the rectangular faces
\[ A_{\text{rectangles}} = A_1 + A_2 + A_3 = 54 + 72 + 90 = 216 \text{ square centimeters} \]
Step 4: Total surface area of the prism
The total surface area \( SA \) of the prism includes the area of the two triangular faces (since there are two identical triangular faces) plus the area of the rectangular faces: \[ SA = 2 \times \text{Area of the triangle} + \text{Area of the rectangles} = 2 \times 24 + 216 \] \[ SA = 48 + 216 = 264 \text{ square centimeters} \]
Therefore, the surface area of the triangular prism is \(\boxed{264}\) square centimeters.