Square A"B"C"D" is the final image after the rule
 was applied to square ABCD.

On a coordinate plane, a square A double-prime B double-prime C double-prime D double-prime has points (negative 5, negative 3), (negative 3, negative 1), (negative 1, negative 3), (negative 3, negative 5).
What are the coordinates of vertex A of square ABCD?

(–1, –6)
(–1, –2)
(–1, 6)
(–2, 1)

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.

On a coordinate plane, a square A double-prime B double-prime C double-prime D double-prime has points (negative 5, negative 3), (negative 3, negative 1), (negative 1, negative 3), (negative 3, negative 5). What are the coordinates of vertex A of square ABCD? (–1, –6) (–1, –2) (–1, 6) (–2, 1)

Solve a coordinate geometry problem involving a square on the plane. Apply the rotation transformation formula to find the original coordinates of square ABCD after a 45-degree rotation. This step-by-step solution is ideal for students in Grades 9-12.

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