The following diagram shows a square where each side has seven dots that divide the side into six equal segments. All the line segments that connect these dots that form a $45^{\circ}$ angle with a side of the square are drawn as shown. The area of the shaded region is 75. Find the area of the original square.

[center][img]https://i.snag.gy/Jzx9Fn.jpg[/img][/center]
1. Let \( x \) be the length of the segment between two adjacent dots on any side of the original square. Since each side of the square is divided into 6 equal segments, the side length of the square is \( 6x \).

2. Each shaded square is formed by connecting dots that are \( x \) units apart along the side of the square, and these connections form a \( 45^\circ \) angle with the side of the square. The side length of each shaded square is \( \frac{x\sqrt{2}}{2} \).

3. The area of each shaded square is:
   \[
   \left( \frac{x\sqrt{2}}{2} \right)^2 = \frac{x^2 \cdot 2}{4} = \frac{x^2}{2}
   \]

4. There are 15 such shaded squares in the diagram. Therefore, the total shaded area is:
   \[
   15 \cdot \frac{x^2}{2} = \frac{15x^2}{2}
   \]

5. Given that the total shaded area is 75, we set up the equation:
   \[
   \frac{15x^2}{2} = 75
   \]

6. Solving for \( x^2 \):
   \[
   15x^2 = 150 \implies x^2 = 10
   \]

7. The area of the original square is:
   \[
   (6x)^2 = 36x^2
   \]

8. Substituting \( x^2 = 10 \) into the area of the square:
   \[
   36x^2 = 36 \cdot 10 = 360
   \]

The final answer is \(\boxed{360}\)