To find the lateral surface area of a triangular prism, we need to calculate the areas of the three rectangular lateral faces.
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Identify the dimensions of the triangular base: The sides of the triangular base are given as 5 cm, 4 cm, and 6 cm. We can denote these sides as \( a = 5 , \text{cm} \), \( b = 4 , \text{cm} \), and \( c = 6 , \text{cm} \).
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Calculate the perimeter of the triangular base: \[ \text{Perimeter} = a + b + c = 5 , \text{cm} + 4 , \text{cm} + 6 , \text{cm} = 15 , \text{cm} \]
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Identify the height of the prism: The height of the prism is given as 12 cm.
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Calculate the lateral surface area: The lateral surface area \( A_L \) of a triangular prism is given by: \[ A_L = \text{Perimeter of the base} \times \text{Height} \] Substituting the values we found: \[ A_L = 15 , \text{cm} \times 12 , \text{cm} = 180 , \text{cm}^2 \]
It seems my initial lateral area calculation gave me 180 \( \text{cm}^2 \), but that is not among the answer choices. The area should be reconsidered with the possibility of misinterpreting the sides provided or misunderstanding the geometry.
The choice of the base triangle needs to be confirmed. The lateral surface includes areas of the rectangular sides corresponding to each side of the triangle multiplied by the height of the prism.
- Expanding the correct rectangle dimensions:
The lengths of the triangular base sides are interpreted as 5 cm, 4 cm, and 6 cm with each side creating a rectangle with the prism height:
- Rectangle corresponding to 5 cm side:
\( 5 , \text{cm} \times 12 , \text{cm} = 60 , \text{cm}^2 \) - Rectangle corresponding to 4 cm side:
\( 4 , \text{cm} \times 12 , \text{cm} = 48 , \text{cm}^2 \) - Rectangle corresponding to 6 cm side:
\( 6 , \text{cm} \times 12 , \text{cm} = 72 , \text{cm}^2 \)
- Rectangle corresponding to 5 cm side:
\[ \text{Total Lateral Surface Area} = 60 + 48 + 72 = 180 , \text{cm}^2 \]
Final Comparison with Given Options: There seems to be an inconsistency or misinterpretation of the problem or units, as none of the answer choices match with the conclusion of this calculation. If only the generated options matter, re-examinations should verify bases about prisms' overall or differential inputs.
However, if there's an implied verticality in dimensional interpretation for additional computation potentially leading toward volume or broader surface scope, without modification, derived continues reflecting at: \[ \text{Correct calculation for given surface availability}: \text{Needs review under selected geometry or surface conditions} \] Given the parameters, the computation and choices as ( 180 , \text{cm}^2 \erupt errors; thus my report assumes misconceiving data structures.
*I would appreciate clarity or adjustments upon dimensions rather than reiterative results set!