To solve the problem, let’s break it down step by step.
1. Plot the Points
The points you have are:
- A (-7, 6)
- B (-2, -6)
- C (10, -1)
- D (5, 11)
You can plot these points on a grid (Cartesian coordinate plane).
2. Connect the Points
To connect these points and form square ABCD, first, we need to determine the correct order of the points to form a square.
- A (-7, 6): This is in the second quadrant.
- B (-2, -6): This is in the fourth quadrant.
- C (10, -1): This is in the fourth quadrant.
- D (5, 11): This is in the second quadrant.
To correctly visualize the square, you'll likely have to rearrange these points. However, properly ordering the points requires calculation of which points are actually the vertices of a square based on distance and slope.
3. Finding the Side Length Using the Pythagorean Theorem
To form a right triangle for calculating the side length, we can use two of the points. For simplicity, let’s use points A and B:
- Calculate the distance between A and B.
\[ \text{Distance (d)} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting A (-7, 6) and B (-2, -6):
- \(x_1 = -7\), \(y_1 = 6\)
- \(x_2 = -2\), \(y_2 = -6\)
\[ d = \sqrt{((-2) - (-7))^2 + ((-6) - 6)^2} \] \[ = \sqrt{(5)^2 + (-12)^2} \] \[ = \sqrt{25 + 144} \] \[ = \sqrt{169} \] \[ = 13 \]
4. Area of Square ABCD
Since we have found the side length \(s = 13\) feet, we can now calculate the area \(A\) of the square.
Using the area formula for a square:
\[ A = s^2 \] \[ A = 13^2 = 169 \text{ square feet} \]
Final Results
- Side length, \(s = 13\) feet.
- Area, \(A = 169\) square feet.
You can now conclude that the side length of square ABCD is 13 feet and the area is 169 square feet.
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