To find the side length \( S \) of square \( ABCD \), we can use the Pythagorean theorem by calculating the distance between two adjacent vertices.
Let's choose vertices A and B for this calculation. You can also use any other adjacent vertices such as B and C, C and D, or D and A. However, for this example, we will find the distance \( AB \).
The coordinates of points A and B are:
- \( A (-7, 6) \)
- \( B (-2, -6) \)
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Plugging in the coordinates of A and B:
\[ d = \sqrt{((-2) - (-7))^2 + ((-6) - 6)^2} \]
Calculating the differences:
- \( (-2) + 7 = 5 \)
- \( (-6) - 6 = -12 \)
Now substitute these values into the distance formula:
\[ d = \sqrt{(5)^2 + (-12)^2} \] \[ d = \sqrt{25 + 144} \] \[ d = \sqrt{169} \] \[ d = 13 \]
Thus, the side length \( S \) of square \( ABCD \) is \( 13 \).