To calculate the surface area of the triangular prism, we need to find the area of the triangular face and the areas of the three rectangular faces.
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Area of the triangular face: Given that the legs of the triangular face are 3 cm and 4 cm, we can use the formula for the area of a right triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \text{ cm} \times 4 \text{ cm} = 6 \text{ cm}^2 \]
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Area of the rectangular faces: The prism has three rectangular faces:
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The first rectangular face (base area), which has a width of 5 cm and a height of 11 cm: \[ \text{Area}_{\text{rect1}} = \text{length} \times \text{width} = 11 \text{ cm} \times 5 \text{ cm} = 55 \text{ cm}^2 \]
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The second rectangular face, which corresponds to the leg of the triangle of 3 cm and the length of the prism (11 cm): \[ \text{Area}_{\text{rect2}} = \text{leg1} \times \text{length} = 3 \text{ cm} \times 11 \text{ cm} = 33 \text{ cm}^2 \]
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The third rectangular face, which corresponds to the leg of the triangle of 4 cm and the length of the prism (11 cm): \[ \text{Area}_{\text{rect3}} = \text{leg2} \times \text{length} = 4 \text{ cm} \times 11 \text{ cm} = 44 \text{ cm}^2 \]
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Total surface area: Now, we can sum the areas of the triangular face and the three rectangular faces:
\[ \text{Total Surface Area} = \text{Area}{\text{triangle}} + \text{Area}{\text{rect1}} + \text{Area}{\text{rect2}} + \text{Area}{\text{rect3}} \]
\[ \text{Total Surface Area} = 6 \text{ cm}^2 + 55 \text{ cm}^2 + 33 \text{ cm}^2 + 44 \text{ cm}^2 = 138 \text{ cm}^2 \]
Thus, the surface area of the triangular prism is 138 square centimeters.