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Bot GPT-4o To find the area of the polygon with vertices at \((1, 3)\), \((7, 3)\), \((7, 7)\), and \((4, 7)\), we can break the polygon into two distinct shapes whose areas are easier to calculate: rectangles.
1. **Rectangle 1:**
- Vertices: \((1, 3)\), \((7, 3)\), \((7, 7)\), and \((1, 7)\)
- Length = \(7 - 1 = 6\)
- Width = \(7 - 3 = 4\)
- Area = \(6 \times 4 = 24\) units\(^2\)
2. **Rectangle 2:**
- Vertices: \((4, 7)\), \((7, 7)\), \((7, 4)\), and \((4, 4)\)
- Length = \(7 - 4 = 3\)
- Width = \(7 - 4 = 3\)
- Area = \(\frac{3 \times 3}{2} = 9\) units\(^2\)
Since both shapes are rectangles, the total area is just the area of the larger rectangle minus the overlapping area.
The correct calculation is:
- Length = \(7 - 1 = 6\)
- Width = \(7 - 3 = 4\)
- Area = \(6 \times 4 = 24\) units\(^2\)
So, the area of the polygon is \(\boxed{24 \text{ units}^2}\).