Solution:
Given, a triangle is to be constructed in the xy-plane such that the x- and y- coordinates of each vertex are integers that satisfy the inequalities -3 x 7 and 2 y 7. Also, one of the sides is parallel to the x-axis.

Finding the range of possible coordinates:
To construct a triangle, we need three vertices. Let’s consider the first vertex, which can have any x-coordinate from -3 to 7 and any y-coordinate from 2 to 7. This gives us a total of (7 - (-3) + 1) * (7 - 2 + 1) = 11 * 6 = 66 possible coordinates for the first vertex.

For the second vertex, we can place it on the x-axis (since one of the sides is parallel to the x-axis). If the first vertex has y-coordinate y1, then the second vertex must have y-coordinate y2 = y1 or y2 = y1 + 1 (since the y-coordinates must be integers). So, there are at most two possible y-coordinates for the second vertex.

If the first vertex has x-coordinate x1, then the second vertex must have x-coordinate x2 such that |x2 - x1| < h,="" where="" h="" is="" the="" length="" of="" the="" horizontal="" side="" of="" the="" triangle.="" since="" the="" horizontal="" side="" is="" parallel="" to="" the="" x-axis,="" h="" is="" the="" difference="" between="" the="" x-coordinates="" of="" the="" first="" and="" second="" vertices.="" so,="" h="x2" -="" x1.="" therefore,="" the="" possible="" values="" of="" x2="" are="" x1="" +="" 1,="" x1="" +="" 2,="" …,="" x1="" +="" h="" -="" />

Now, we need to find the possible values of x2 for each possible value of x1. If x1 = -3, then h can be 1, 2, 3, 4, 5, or 6. If x1 = -2, then h can be 1, 2, 3, 4, 5, 6, or 7, but we need to exclude the case where h = 7 because that would make the triangle degenerate (i.e., it would be a straight line). Similarly, if x1 = -1, then h can be 1, 2, 3, 4, 5, or 6, but we need to exclude the case where h = 6. For x1 = 0, 1, 2, 3, 4, 5, 6, we need to exclude the cases where h is equal to the length of the horizontal side of the triangle.

Counting the number of triangles:
For each possible pair of vertices (i.e., first and second vertices), we can count the number of possible third vertices. Since the third vertex cannot lie on the x-axis (otherwise, the triangle would be degenerate), the y-coordinate of the third vertex must be either y1 or y1 + 1 (depending on whether the second vertex has y-coordinate y1 or y1 + 1). If the second vertex has x-coordinate x2, then the possible x-coordinates of the third vertex are the integers between x1 and x2 that are not equal to x1 or x2. Therefore, the number of possible third vertices is (|x2 - x1| -