To find the square's area and side length using the given points, we first need to identify the structure of the square formed by these points. However, based on the coordinates provided, \( (-7, 6) \), \( (-2, -6) \), \( (10, -1) \), and \( (5, 11) \), it is not immediately clear that these points form a square.
To check, we will calculate the pairwise distances between the points using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate distances between pairs of points
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Distance between \((-7, 6)\) and \((-2, -6)\): \[ d_1 = \sqrt{((-2) - (-7))^2 + ((-6) - 6)^2} = \sqrt{(5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
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Distance between \((-7, 6)\) and \( (10, -1)\): \[ d_2 = \sqrt{(10 - (-7))^2 + (-1 - 6)^2} = \sqrt{(17)^2 + (-7)^2} = \sqrt{289 + 49} = \sqrt{338} \approx 18.38 \]
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Distance between \((-7, 6)\) and \( (5, 11)\): \[ d_3 = \sqrt{(5 - (-7))^2 + (11 - 6)^2} = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
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Distance between \((-2, -6)\) and \( (10, -1)\): \[ d_4 = \sqrt{(10 - (-2))^2 + ((-1) - (-6))^2} = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
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Distance between \((-2, -6)\) and \( (5, 11)\): \[ d_5 = \sqrt{(5 - (-2))^2 + (11 - (-6))^2} = \sqrt{(7)^2 + (17)^2} = \sqrt{49 + 289} = \sqrt{338} \approx 18.38 \]
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Distance between \( (10, -1)\) and \( (5, 11)\): \[ d_6 = \sqrt{(5 - 10)^2 + (11 - (-1))^2} = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
Step 2: Analyze the distances
From our calculations:
- \( d_1 = 13 \)
- \( d_3 = 13 \)
- \( d_4 = 13 \)
- \( d_6 = 13 \)
- \( d_2 \approx 18.38 \)
- \( d_5 \approx 18.38 \)
Since four of these distances are equal (13) and the other two are equal to each other but different (approximately 18.38), it indicates that we do not have a square, but we might have a rectangle instead.
Step 3: Determine the area of the shape formed
If the points are vertices of a rectangle, the lengths of the sides can be determined from the distances calculated above.
Let’s assume the shorter sides are of length 13 and the longer sides are of length \(13\) and \(18.38\).
The area \(A\) of the rectangle can thus be calculated as:
\[ A = \text{length} \times \text{width} = 13 \times 13 = 169 \]
However, we must recognize that we are calculating based on the observations above, we would not derive a square here. The corresponding square dimensions would be for side length \(s\) as required, and that length is \(13\).
Final Answer
Side Length: \(13\)
Area: \(169\)