To determine the coordinates and area of rectangle ABCD, let's first visualize the vertices based on the provided coordinates:
- Point A: (-3, -2)
- Point B: (4, -2)
- Point C: (4, 1)
- Point D: (-3, 1)
Next, let's check if these points indeed form a rectangle. A rectangle must have opposite sides equal in length and all angles equal to 90 degrees.
Vertices and Lengths:
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AB: The distance between points A and B can be calculated as: \[ AB = |x_B - x_A| = |4 - (-3)| = |4 + 3| = 7 \]
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BC: The distance between points B and C is: \[ BC = |y_C - y_B| = |1 - (-2)| = |1 + 2| = 3 \]
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CD: The distance between points C and D is: \[ CD = |x_D - x_C| = |-3 - 4| = |-7| = 7 \]
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DA: The distance between points D and A is: \[ DA = |y_A - y_D| = |-2 - 1| = |-3| = 3 \]
Checking Rectangle Properties:
- We see that opposite sides are equal: \( AB = CD = 7 \) and \( BC = DA = 3 \).
- Since there are two pairs of equal opposite sides, and the angles between these sides are 90 degrees (given the coordinate system), it confirms that ABCD forms a rectangle.
Area Calculation:
The area \( A \) of a rectangle can be computed using the formula: \[ \text{Area} = \text{length} \times \text{width} \]
From our calculations:
- Length (AB or CD) = 7 units
- Width (BC or DA) = 3 units
Thus, the area is: \[ \text{Area} = 7 \times 3 = 21 \text{ square units} \]
Therefore, the area of rectangle ABCD is \(\boxed{21}\) units square.
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