To find the area of the rectangle defined by the coordinates A(0,0), B(4,-4), C(7,-1), and D(3,3), we first need to confirm whether these points indeed form a rectangle, and then we can calculate the area.
Step 1: Verify Coordinates Form a Rectangle
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Identify Vertices:
- A(0, 0)
- B(4, -4)
- C(7, -1)
- D(3, 3)
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Check Distances to Determine Perpendicular Sides:
- Calculate lengths of the sides:
- AB = √[(4-0)² + (-4-0)²] = √[4² + (-4)²] = √[16 + 16] = √32 = 4√2
- BC = √[(7-4)² + (-1+4)²] = √[3² + (3)²] = √[9 + 9] = √18 = 3√2
- CD = √[(7-3)² + (-1-3)²] = √[4² + (-4)²] = √[16 + 16] = √32 = 4√2
- DA = √[(3-0)² + (3-0)²] = √[3² + 3²] = √[9 + 9] = √18 = 3√2
- Calculate lengths of the sides:
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Check Angles to Verify Right Angles:
- Using the slope formula, we find:
- Slope of AB = (-4-0)/(4-0) = -1
- Slope of BC = (-1-(-4))/(7-4) = 3/3 = 1
- Slope of CD = (3-(-1))/(3-7) = 4/(-4) = -1
- Slope of DA = (3-0)/(3-0) = 1
- Using the slope formula, we find:
Since opposite sides are equal and slopes of consecutive sides are negative reciprocals, we conclude these points form a rectangle.
Step 2: Calculate the Area
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Dimensions of the Rectangle:
- Length (along AB or CD) = 4√2
- Width (along BC or DA) = 3√2
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Area of the Rectangle: \[ \text{Area} = \text{Length} \times \text{Width} = (4\sqrt{2}) \times (3\sqrt{2}) \] \[ \text{Area} = 12 \times 2 = 24 \]
Conclusion
Hence, the exact area of the rectangle is \(\boxed{24}\) square units.