find the lattice point for a rectangle whose lenght is 3 times its width and no sides are horizontal

Answer 1

Let's let our rectangle be ABCD. If you can't have the sides be horizontal, let's make the diagonal horizontal and place one corner of the rectangle on the origin (0, 0):

Considering the large red triangle (which is half of our rectangle), we can say tan(θ) = 3w/w or tan(θ) = 3.

For the smaller triangle ABE, we can say tan(θ) = y/x where "y" is the y-coordinate of point B and "x" is the x-coordinate of point B. So, since tan(θ) = 3 and tan(θ) = y/x then y/x = 3.

If we arbitrarily select x = 1, then y = 3 so the coordinate of point B is (1, 3). You can use the Pythagorean Theorem to find the length of AB as √10. That means the length of BC is 3√10 and, again using the Pythagorean Theorem, the length of AC is 10 which makes point C at (10, 0). You can then solve for the coordinates of point D, which would be (9, -3).

Of course, since we arbitrarily selected x = 1, this rectangle can be scaled up by selecting x = 2 or any other integer and scaling the other coordinates in concert with this.