Shortest and longest XY

You construct a rectangle ABCD with AB = 7 cm and BC = 4 cm. So AB and CD are the long sides, 7 cm each, and BC and AD are the short sides, 4 cm each, facing each other across the rectangle. Now imagine a point X that can slide anywhere along the side AD, even onto A or D, and a point Y that can slide anywhere along the side BC, even onto B or C.

When are X and Y closest together? Try X 5 mm from A and Y 5 mm from B; then X 1 cm from A and Y 1 cm from B; then 1 cm 5 mm from each. Every time, when X and Y are the same distance from A and B, the line XY runs straight across, parallel to AB, and XY = AB = 7 cm. The four-sided figure ABYX is then a rectangle, so its opposite sides XY and AB are equal. This is the shortest XY can ever be: X and Y can never be closer than 7 cm, because the two short sides are 7 cm apart.

If X and Y are at different distances from A and B, XY slants, and a slanting line across the rectangle is longer than 7 cm.

When are they farthest apart? When XY slants as much as it can, from one corner to the opposite corner. That happens when X is at A and Y is at C, so XY is the diagonal AC, or when X is at D and Y is at B, so XY is the diagonal BD. No line inside the rectangle is longer than a diagonal.