To proceed with your request, we need to first check if the points given form a square and then calculate the relevant properties if they do.
The points are:
- A (-7, 6)
- B (-2, -6)
- C (10, -1)
- D (5, 11)
Let's first find the distances between each pair of points to identify the sides and diagonals:
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Calculate the distances:
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]-
Distance \( AB \): \[ d_{AB} = \sqrt{(-2 - (-7))^2 + (-6 - 6)^2} = \sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
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Distance \( BC \): \[ d_{BC} = \sqrt{(10 - (-2))^2 + (-1 - (-6))^2} = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
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Distance \( CD \): \[ d_{CD} = \sqrt{(5 - 10)^2 + (11 - (-1))^2} = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
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Distance \( DA \): \[ d_{DA} = \sqrt{(-7 - 5)^2 + (6 - 11)^2} = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
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Check Diagonal lengths:
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Distance \( AC \): \[ d_{AC} = \sqrt{(10 - (-7))^2 + (-1 - 6)^2} = \sqrt{(17)^2 + (-7)^2} = \sqrt{289 + 49} = \sqrt{338} \]
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Distance \( BD \): \[ d_{BD} = \sqrt{(5 - (-2))^2 + (11 - (-6))^2} = \sqrt{(7)^2 + (17)^2} = \sqrt{49 + 289} = \sqrt{338} \]
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Establishing characteristics of ABCD:
- All four sides are equal (13), and the diagonals are equal (\(\sqrt{338}\)), which indicates that ABCD is a square.
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Finding the area:
- The side length \( s \) is \( 13 \).
- Area \( A \) can be calculated as: \[ A = s^2 = 13^2 = 169 \]
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Result summary:
- Side length, \( s \): \( 13 \)
- Area, \( A \): \( 169 \)
To visualize the square, plot the points A, B, C, and D on a coordinate grid. The square would have vertices at those coordinates, with the connections forming the sides and confirming it meets the properties described above.