2019 AMC 10 B

Notice that whatever point we pick for , will be the base of the triangle. Without loss of generality, let points and be and , since for any other combination of points, we can just rotate the plane to make them and under a new coordinate system. When we pick point , we have to make sure that its coordinate is , because that's the only way the area of the triangle can be .

Now when the perimeter is minimized, by symmetry, we put in the middle, at .  We can easily see that and will both be . The perimeter of this minimal tiangle is . which is larger than . Since the minimum perimeter is greater than , there is no triangle that satisfies the condition, giving us .

Without loss of generality, let be a horizontal segment of length . Now realize that has to lie on one of the lines parallel to and vertically units away from it. But is already , and this doesn't form a triangle. Otherwise, without loss of generality,  . Dropping altitude , we have a right triangle with hypotenuse and leg , which is clearly impossible, again giving the answer as .