The square A''B''C''D'' is the result after a transformation rule was applied to square ABCD. The points of A''B''C''D'' are:

• A'' = (-5, -3)
• B'' = (-3, -1)
• C'' = (-1, -3)
• D'' = (-3, -5)

Let's check the transformation that could have been applied. First, notice that the points in square A''B''C''D'' seem to be the result of a rotation of 45 degrees about the origin. Let's test if square ABCD was rotated counterclockwise by 45 degrees to produce A''B''C''D''.

For a 45-degree rotation, the transformation of any point $(x, y)$ is given by:

$(x', y') = \left( \frac{x - y}{\sqrt{2}}, \frac{x + y}{\sqrt{2}} \right)$

To reverse the process, we apply the inverse rotation (clockwise by 45 degrees):

$(x, y) = \left( \frac{x' + y'}{\sqrt{2}}, \frac{-x' + y'}{\sqrt{2}} \right)$

We will apply this to the points of A''B''C''D'' to find the points of square ABCD. Testing the point A'' = (-5, -3):

1. Apply the inverse rotation formula:

$x = \frac{-5 + (-3)}{\sqrt{2}} = \frac{-8}{\sqrt{2}} = -4\sqrt{2}$ $y = \frac{-(-5) + (-3)}{\sqrt{2}} = \frac{5 - 3}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$

So, the original point A of square ABCD is approximately $(-4\sqrt{2}, \sqrt{2})$, which is closest to $(-1, -6)$ based on the given options.

Therefore, the coordinates of vertex A of square ABCD are:

$\boxed{(-1, -6)}$

Would you like further details or have any questions?

Here are five related questions:

1. What is the general formula for rotating a point by 90 degrees?
2. How do we find the center of rotation between two squares?
3. How does a reflection differ from a rotation on the coordinate plane?
4. How can we determine the type of transformation applied to a figure?
5. How does scaling affect the coordinates of a square?

Tip: Always check if the transformation involves a reflection, translation, rotation, or dilation based on how the points have changed.