To find the area of the polygon, we will calculate the areas of different sections of the figure step by step.
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Area of the base pillars: Each pillar is a rectangle with dimensions 4 (length) and 3 (width). \[ \text{Area of one pillar} = \text{length} \times \text{width} = 4 \times 3 = 12 , \text{square units} \] Since there are two pillars: \[ \text{Area of two pillars} = 2 \times 12 = 24 , \text{square units} \]
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Area of the door: The door is a rectangle with a width of 2 and a height of 3.5. \[ \text{Area of the door} = \text{width} \times \text{height} = 2 \times 3.5 = 7 , \text{square units} \]
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Area of the triangular roof: The base of the triangle is the total distance across the house at the base of the roof (which includes the extensions from the pillars) plus the width of the door.
- The base of the triangle consists of the widths of the pillars plus the additional width on either side: \[ \text{Base of the triangle} = 4 + 2 + 4 = 10 , \text{units} \] The height of the triangle is given as 6. \[ \text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 6 = 30 , \text{square units} \]
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Area of the chimney: The chimney is a rectangle with a width of 1 and a height of 2. \[ \text{Area of the chimney} = \text{width} \times \text{height} = 1 \times 2 = 2 , \text{square units} \]
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Total area of the polygon: Finally, we add up the areas of the pillars, door, roof, and chimney: \[ \text{Total area} = \text{Area of pillars} + \text{Area of door} + \text{Area of triangle} + \text{Area of chimney} \] \[ \text{Total area} = 24 + 7 + 30 + 2 = 63 , \text{square units} \]
Since 63 square units is not among the options listed, it’s possible we might have made an error or misunderstood certain dimensions.
After checking the components, if we assume that the door and chimney are embedded in the overall structure and not taking away from the area, the discrepancy must come from proper calculations rather than the interpretation.
Based on the above calculations: Response: None of the options provided is correct as the area computes to 63 square units. Please double-check the problem setup and dimensions given in the problem statement to ensure accuracy.