We are given that the figure $ABCD$ is a square, and we are provided with the coordinates of points $A(3, 5)$ and $D(10, 5)$. The problem is to find the coordinates of point $C$.

Step-by-step analysis:

1. Coordinates of $A$ and $D$:

   • $A(3, 5)$
   • $D(10, 5)$

   These two points lie on a horizontal line, as they share the same $y$-coordinate, meaning the side $AD$ is parallel to the $x$-axis.

2. Length of side $AD$: Since $ABCD$ is a square, the length of each side is the same. The distance between $A$ and $D$ (which is the side length of the square) is the difference in their $x$-coordinates: $\text{Distance} = 10 - 3 = 7$

3. Coordinates of $B$: Point $B$ is directly above point $A$ since $AB$ is vertical, and the distance between $A$ and $B$ is equal to the side length of the square (7). Therefore, the $x$-coordinate of $B$ will be the same as $A$'s, and its $y$-coordinate will be: $B(3, 5 + 7) = B(3, 12)$

4. Coordinates of $C$: Since $BC$ is horizontal, the $y$-coordinate of $C$ will be the same as that of $B$, and its $x$-coordinate will be the same as $D$'s. Therefore, the coordinates of $C$ will be: $C(10, 12)$

Thus, the coordinates of point $C$ are $\boxed{(10, 12)}$.

Would you like more details on this solution or have any questions?

Here are five related questions to expand your understanding:

1. How would the coordinates change if $ABCD$ was not a square but a rectangle?
2. What is the perimeter of the square $ABCD$?
3. How can you find the area of square $ABCD$?
4. What are the coordinates of the center of square $ABCD$?
5. How would you determine the slope of line $AC$?

Tip: When working with geometric figures on a coordinate plane, using the properties of shapes (like parallel sides and equal lengths) can simplify finding unknown coordinates.