2017 AMC 10 A

Because , , , and are lattice points, there are only a few coordinates that actually satisfy the equation. The coordinates are , , , and. We want to maximize and minimize . They also have to be the square root of something, because they are both irrational. The greatest value of happens when and are almost directly across from each other and are in different quadrants. For example, the endpoints of the segment could be and because the two points are almost across from each other. Another possible pair could be and . To find out which segment is longer, we have to compare the distances from their endpoints to a diameter (which must be the longest possible segment). The closest diameter would be from to . The distance between and is shorter than the distance between and Therefore, the segment from to is the longest attainable. The least value of is when the two endpoints are in the same quadrant and are very close to each other. This can occur when, for example, is and is . They are in the same quadrant and no other point on the circle with integer coordinates is closer to the point than . Using the distance formula, we get that is and that is . .