To solve the problem, we need to find all possible rectangles $ABCD$ where:

1. Side $AB$ is parallel to the $x$-axis.
2. The coordinates of the vertices of the rectangle must satisfy:
   • For $x$: $-6 \leq x \leq 6$
   • For $y$: $5 \leq y \leq 15$
3. Both $x$ and $y$ must be integers.

Understanding the Setup

The rectangle is defined by its vertices $A(x_1, y_1)$, $B(x_2, y_1)$, $C(x_2, y_2)$, and $D(x_1, y_2)$, where:

• $A(x_1, y_1)$ and $B(x_2, y_1)$ lie on the same horizontal line (i.e., they have the same $y_1$).
• $C(x_2, y_2)$ and $D(x_1, y_2)$ lie on another horizontal line (i.e., they have the same $y_2$).

Steps to find all possible rectangles:

1. Possible $x$-coordinates: Since $-6 \leq x \leq 6$ and $x$ must be an integer, the possible values for $x_1$ and $x_2$ are: $x_1, x_2 \in \{-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6\}$ There are 13 possible $x$-values.

2. Possible $y$-coordinates: Since $5 \leq y \leq 15$ and $y$ must be an integer, the possible values for $y_1$ and $y_2$ are: $y_1, y_2 \in \{5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}$ There are 11 possible $y$-values.

Counting the Possible Rectangles

• For the rectangle to exist, $x_1 \neq x_2$ (i.e., the $x$-coordinates of points $A$ and $B$ should be different).
• Similarly, $y_1 \neq y_2$ (i.e., the $y$-coordinates of points $A$ and $D$ should be different).

1. Choosing distinct $x$-coordinates: From the 13 possible $x$-values, there are $\binom{13}{2} = \frac{13 \times 12}{2} = 78$ ways to choose two distinct $x$-coordinates.

2. Choosing distinct $y$-coordinates: From the 11 possible $y$-values, there are $\binom{11}{2} = \frac{11 \times 10}{2} = 55$ ways to choose two distinct $y$-coordinates.

Total Number of Rectangles

The total number of possible rectangles is the product of the two combinations: $78 \times 55 = 4290$

Thus, there are 4290 possible rectangles that can be constructed under the given conditions.

Would you like further details on the steps or concepts involved? Here are five related questions for you to consider:

1. How would the problem change if the side $AB$ were not constrained to be parallel to the $x$-axis?
2. How does the choice of distinct $x_1$ and $x_2$ affect the possible dimensions of the rectangle?
3. What if the constraint on $y$-values changed to $0 \leq y \leq 10$?
4. Could we solve a similar problem for a triangle instead of a rectangle? How would it differ?
5. How does the symmetry of the coordinate plane simplify solving this problem?

Tip: For combinatorics problems involving distinct coordinates, always check if there are constraints like parallelism or axis alignment!