To determine the coordinates of the rectangle given points A, B, and C, let's start by identifying the locations of points A, B, and C.
- Coordinates of the points:
- A = (-3, -1)
- B = (-3, 0)
- C = (3, 4)
Next, we will find the coordinates of point D, which completes the rectangle.
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Finding the fourth point (D):
- Since A and B have the same x-coordinate, they form a vertical side of the rectangle.
- C is the diagonally opposite point from A, and thus point D can be determined using the x-coordinate of C and the y-coordinate of B for D.
Therefore, D should have the coordinates:
- D = (3, -1)
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Coordinates of the rectangle's corners:
- A (-3, -1)
- B (-3, 0)
- C (3, 4)
- D (3, -1)
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Calculating the lengths of sides:
- The length of AB = distance between A and B (vertical side): \[ |y_B - y_A| = |0 - (-1)| = 1 \text{ unit} \]
- The length of AD = distance between A and D (horizontal side): \[ |x_D - x_A| = |3 - (-3)| = 6 \text{ units} \]
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Compute the area of the rectangle:
- Area = length × width \[ \text{Area} = 1 , \text{unit} \times 6 , \text{units} = 6 , \text{units}^2 \]
However, I made a mistake on the calculation of the lengths of sides. Let's compute the right dimensions instead.
- Length of side AB is indeed 1 unit.
- Length of side BC can be calculated:
- For side BC, the vertical distance between the y-coordinates of B and C is:
- \( |y_C - y_B| = |4 - 0| = 4 \text{ units} \)
- For side BC, the vertical distance between the y-coordinates of B and C is:
Therefore, the rectangle's dimensions are correctly identified as follows:
- Width = AC = 6 units
- Height = 4 units
Thus, the area: \[ \text{Area} = 6 , \text{units} \times 4 , \text{units} = 24 , \text{units}^2 \]
After reviewing all properties of the points given, NONE of the responses for possible areas match our calculations. Hence, the area of our rectangle formed by points A, B, and C is indeed: \[ \text{Area} = 24 \text{ square units} \]