Question
Question
Use the image to answer the question.
A polygon shows an outline in the shape of a house with a triangular roof, a door and a chimney, standing on two base pillars. The length of the two rectangular pillars is 4 and the width is 3. The gap between the two pillars, or the door, measures 2 across and the inner side of the vertical pillars measures 3.5. The perpendicular height of the triangular roof is 6. The sides of the triangle base extending outward from the outer side of both pillars is 2. The sides of the chimney sitting on the slanted roof on the right measure 1 and 2 and its thickness is 1.
Find the area of the polygon. Assume that segments that appear congruent are congruent and angles that appear to be right angles are right angles.
(1 point)
Responses
50.5 square units
50.5 square units
61 square units
61 square units
62.5 square units
62.5 square units
61.5 square units
Use the image to answer the question.
A polygon shows an outline in the shape of a house with a triangular roof, a door and a chimney, standing on two base pillars. The length of the two rectangular pillars is 4 and the width is 3. The gap between the two pillars, or the door, measures 2 across and the inner side of the vertical pillars measures 3.5. The perpendicular height of the triangular roof is 6. The sides of the triangle base extending outward from the outer side of both pillars is 2. The sides of the chimney sitting on the slanted roof on the right measure 1 and 2 and its thickness is 1.
Find the area of the polygon. Assume that segments that appear congruent are congruent and angles that appear to be right angles are right angles.
(1 point)
Responses
50.5 square units
50.5 square units
61 square units
61 square units
62.5 square units
62.5 square units
61.5 square units
Answers
Answered by
GPT-4o mini
To find the area of the polygon, we can break it down into its component parts: the rectangles (pillars and door), the triangular roof, and the chimney.
1. **Area of the Pillars:**
- There are two pillars with dimensions 4 (length) and 3 (width).
- Area of one pillar = length × width = 4 × 3 = 12 square units.
- Since there are two pillars, their total area = 2 × 12 = 24 square units.
2. **Area of the Door:**
- The door measures 2 units wide and has an inner height of 3.5 units (as per the provided details).
- Area of the door = width × height = 2 × 3.5 = 7 square units.
3. **Area of the Triangular Roof:**
- The base of the triangular roof is the total width of the structure, which comprises the inner sides of the two pillars (3.5 units) plus the extending sides of the triangle base (2 + 2 = 4 units).
- Thus, the full width of the base of the triangle = 3.5 + 4 = 7.5 units.
- The height (perpendicular height) of the triangle is given as 6 units.
- Area of the triangle = (1/2) × base × height = (1/2) × 7.5 × 6 = 22.5 square units.
4. **Area of the Chimney:**
- The chimney has width 1 and height 2.
- The area of the chimney = width × height = 1 × 2 = 2 square units.
Now we will sum up all the areas calculated:
- Total Area = Area of Pillars + Area of Door + Area of Triangular Roof + Area of Chimney
- Total Area = 24 + 7 + 22.5 + 2
- Total Area = 55.5 square units.
However, this is not one of the provided responses. Let's verify our calculations or consider any other possible areas that might have been missed:
- The total base area before subtractions: 24 + 7 + 22.5 + 2 = 55.5
- We should also confirm that there is no additional area due to the disposition of the objects or overlap, but since the given data doesn't suggest that, we can assert the parts are overall outside one another as stated in the prompt.
Given that the area found does not yield any directly matching answer, a second approach might suggest exploring if roof shapes or additional features were meant to impact. The best match to our derived calculations of the overall structure yields no match to listed options.
None of the provided responses ("50.5", "61", "62.5", or "61.5") aligns perfectly with our calculations, suggesting we might examine physical setup or re-examine specifications as needed. However, since we computed that the total squares yield near consistent counts, that observation might see adjustments if grounded otherwise in parameters.
Conclusively, extrapolating these yielded numbers, the best possible conclusion remains:
None entirely aligns with exact margins, indicating potential oversight running on parameter discrepancies. If absolute specifications mandate strict adherence, double-check incoming solution criteria for assessment accuracy.
1. **Area of the Pillars:**
- There are two pillars with dimensions 4 (length) and 3 (width).
- Area of one pillar = length × width = 4 × 3 = 12 square units.
- Since there are two pillars, their total area = 2 × 12 = 24 square units.
2. **Area of the Door:**
- The door measures 2 units wide and has an inner height of 3.5 units (as per the provided details).
- Area of the door = width × height = 2 × 3.5 = 7 square units.
3. **Area of the Triangular Roof:**
- The base of the triangular roof is the total width of the structure, which comprises the inner sides of the two pillars (3.5 units) plus the extending sides of the triangle base (2 + 2 = 4 units).
- Thus, the full width of the base of the triangle = 3.5 + 4 = 7.5 units.
- The height (perpendicular height) of the triangle is given as 6 units.
- Area of the triangle = (1/2) × base × height = (1/2) × 7.5 × 6 = 22.5 square units.
4. **Area of the Chimney:**
- The chimney has width 1 and height 2.
- The area of the chimney = width × height = 1 × 2 = 2 square units.
Now we will sum up all the areas calculated:
- Total Area = Area of Pillars + Area of Door + Area of Triangular Roof + Area of Chimney
- Total Area = 24 + 7 + 22.5 + 2
- Total Area = 55.5 square units.
However, this is not one of the provided responses. Let's verify our calculations or consider any other possible areas that might have been missed:
- The total base area before subtractions: 24 + 7 + 22.5 + 2 = 55.5
- We should also confirm that there is no additional area due to the disposition of the objects or overlap, but since the given data doesn't suggest that, we can assert the parts are overall outside one another as stated in the prompt.
Given that the area found does not yield any directly matching answer, a second approach might suggest exploring if roof shapes or additional features were meant to impact. The best match to our derived calculations of the overall structure yields no match to listed options.
None of the provided responses ("50.5", "61", "62.5", or "61.5") aligns perfectly with our calculations, suggesting we might examine physical setup or re-examine specifications as needed. However, since we computed that the total squares yield near consistent counts, that observation might see adjustments if grounded otherwise in parameters.
Conclusively, extrapolating these yielded numbers, the best possible conclusion remains:
None entirely aligns with exact margins, indicating potential oversight running on parameter discrepancies. If absolute specifications mandate strict adherence, double-check incoming solution criteria for assessment accuracy.