To find the perimeter of the rectangle defined by the given points, we first need to determine the lengths of its sides.
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Identify the coordinates given:
- A(-3, 1)
- B(3, 4)
- C(7, 2)
- D(1, 1)
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Find the lengths of the sides of the rectangle. The perimeter \(P\) of a rectangle can be calculated using the formula: \[ P = 2 \times (\text{length} + \text{width}) \]
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To determine the vertices that make up the rectangle, we need to find the distance between points. We will calculate the distances between pairs of points to identify the lengths of the sides.
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Calculate the distance between each pair of points:
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Distance between A(-3, 1) and B(3, 4): \[ d_{AB} = \sqrt{(3 - (-3))^2 + (4 - 1)^2} = \sqrt{(3 + 3)^2 + (4 - 1)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \]
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Distance between B(3, 4) and C(7, 2): \[ d_{BC} = \sqrt{(7 - 3)^2 + (2 - 4)^2} = \sqrt{(4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]
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Distance between C(7, 2) and D(1, 1): \[ d_{CD} = \sqrt{(1 - 7)^2 + (1 - 2)^2} = \sqrt{(-6)^2 + (-1)^2} = \sqrt{36 + 1} = \sqrt{37} \]
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Distance between D(1, 1) and A(-3, 1): \[ d_{DA} = \sqrt{(-3 - 1)^2 + (1 - 1)^2} = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4 \]
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- The dimensions of the rectangle might be understood from the calculations. Assuming one side is 4 units and the other is some length determined by the other points, we can look back to see the distances we calculated.
However, none of the given lengths seem to pair up to create the standard rectangle. This leads us to believe the question may be improperly interpreting the figure.
Assuming a rectangle created:
- width = DC = 4
- length = AB = 3\sqrt{5}
- Finally, calculate the perimeter: \[ P = 2(4 + 3\sqrt{5}) = 8 + 6\sqrt{5} \text{ "inches"} \]
There is no option provided that exactly matches this expression.
Please double-check the data or details regarding the rectangle and check coordinate pairs against the figure illustrated if available. From the distances calculated, the perimeter cannot be simplified to any of the provided responses explicitly.
None of the responses match, but if you average a rough approximation around "105–√ in", please check if it fits as the side lengths precisely.
5 answers