We are given a pattern of four identical squares, with the sides parallel to the axes, and need to determine the coordinates of point $C$.

Key Information:

• Point $A$ has coordinates $(6, 7)$.
• Point $B$ has coordinates $(34, 32)$.
• The pattern consists of four squares of equal size.
• The task is to determine the coordinates of point $C$.

Approach:

1. First, find the side length of each square. The total horizontal distance between points $A$ and $B$ is: $\Delta x = 34 - 6 = 28$ The total vertical distance is: $\Delta y = 32 - 7 = 25$ Since there are four squares, the length of each square's side is the greatest common divisor (GCD) of 28 and 25. Since 28 and 25 are co-prime, the side length is the same as the smallest of these distances. However, we can assume from symmetry and the diagram that the horizontal and vertical distances are divided equally among the four squares.

2. Each square's side length can be approximated by dividing the total horizontal and vertical distances by 4: $\text{Side length} = \frac{28}{4} = 7 \quad \text{and} \quad \frac{25}{4} = 6.25$ Therefore, each square appears to have a side length close to 7 in both directions, with slight adjustments for vertical distance.

3. Now, point $C$ lies on the boundary of the second square, after moving horizontally and vertically from point $A$. Since point $C$ is the vertex between the second and third squares, it will be displaced horizontally by two square widths and vertically by two square heights.

4. Moving from $A(6, 7)$:

   • Horizontally by $2 \times 7 = 14$: $6 + 14 = 20$
   • Vertically by $2 \times 6.25 \approx 12.5$: $7 + 12.5 = 19.5$

Thus, the coordinates of point $C$ are approximately $(20, 19.5)$.

Would you like further details or clarifications on the steps? Here are some related questions you might find helpful:

1. How do we verify the side lengths of the squares more rigorously?
2. Why is the greatest common divisor (GCD) not relevant in this case?
3. How do we apply transformations in geometry to solve similar problems?
4. How do coordinate shifts work in a geometric pattern like this?
5. Can we solve for the exact side length using another method?

Tip: When dealing with geometric patterns, always break down the pattern into symmetrical units (e.g., squares) to make calculations easier.