To find the surface area of the triangular pyramid, we need to calculate the area of the triangular base and the area of the three triangular faces.
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Area of the base (a triangle): The formula for the area of a triangle is: \[ \text{Area}{base} = \frac{1}{2} \times \text{base} \times \text{height} \] Here, the base is 10 and the height is 8.7. \[ \text{Area}{base} = \frac{1}{2} \times 10 \times 8.7 = 43.5 \]
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Area of the lateral faces: Each lateral face is a triangle with a base of the side of the triangular base (which is 10 for the given problem) and a height of the slant height (12). Since all sides are not provided, and considering a equilateral triangle for this case where each side is equal to the base of 10, we calculate the area of one triangular face first: \[ \text{Area}{face} = \frac{1}{2} \times \text{base} \times \text{slant height} \] Using a side of 10 and slant height of 12: \[ \text{Area}{face} = \frac{1}{2} \times 10 \times 12 = 60 \] Since there are three triangular faces, multiply by 3: \[ \text{Total Area of faces} = 3 \times 60 = 180 \]
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Total surface area: Now, add the area of the base to the total area of the faces: \[ \text{Total Surface Area} = \text{Area}_{base} + \text{Total Area of faces} \] \[ \text{Total Surface Area} = 43.5 + 180 = 223.5 \]
Thus, the surface area of the triangular pyramid is 223.5. So the correct response is:
223.5