To find the perimeter of rectangle ABCD where point A is at (-1, -1) and point B is at (1, 2), and the length of diagonal BD is 6 units, we can follow these steps:
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Identify Points and Diagonal: Let’s assign points based on the rectangle:
- A = (-1, -1)
- B = (1, 2)
We need to find points C and D such that B and D form one diagonal of the rectangle, and A and C form the other.
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Calculate the Length of the Diagonal: The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Since we have point B and we are assuming point D as \((x, y)\), we know: \[ \sqrt{(x - 1)^2 + (y - 2)^2} = 6 \]
Squaring both sides gives us: \[ (x - 1)^2 + (y - 2)^2 = 36 \]
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Finding Point D: Also, note that since A and B are opposite corners of the rectangle, we can use the coordinates of A to help find D. If C is directly across from A, it can be determined that C = (x_c, y_c) must also satisfy the rectangle configuration, assuming proper alignment.
Knowing the coordinates: \(y_B - y_A = 2 - (-1) = 3\) (the length in the y-axis) \(x_B - x_A = 1 - (-1) = 2\) (the length in the x-axis)
This means length AB = 3 and length AD = 2.
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Calculating Rectangle Perimeter: The perimeter \(P\) of a rectangle can be calculated as: \[ P = 2 \times (length + width) \] where \(length = 3\) (vertical distance) and \(width = 2\) (horizontal distance).
Therefore, the perimeter is: \[ P = 2 \times (3 + 2) = 2 \times 5 = 10 \]
Hence, the perimeter of rectangle ABCD is:
\[ \text{Perimeter} = 10.0 \]
Rounding to the nearest tenth, the final answer is \(10.0\).