Open Middle®
Directions: Using the digits 1 to 8, at most one time each, fill in the coordinates to create the vertices of a rectangle: A(__, __), B(__, __), C(__, __), D(__, __).
Extension: What is the rectangle with the largest/smallest area/perimeter that you can find?
Hint
Which lattice points can be used to create the rectangle? Which lattice points cannot be used?
Answer
The Largest Rectangle: area 16, perimeter 12sqrt(2)
(1,4)(3,2)(7,6)(5,8)
(1,6)(3,8)(7,4)(5,2)
The Largest Square: area 20, perimeter 8sqrt(5)
(1,6)(5,8)(7,4)(3,2)
(1,4)(3,8)(7,6)(5,2)
The Smallest Rectangle: area 4, perimeter 6sqrt(2)
(1,7)(2,8)(4,6)(3,5)
(5,2)(7,4)(8,3)(6,1)
The Smallest Square: area 5, perimeter 4sqrt(5)
(1,7)(2,5)(4,6)(3,8)
(1,6)(2,8)(3,5)(4,7)
For each of the rectangles above, an additional rectangle can be created by swapping the x and y coordinates (the reflection about line y=x)
Source: Erick Lee
I have a doubt. In the “Solution” section you say that:
* the smallest square has area 17 but I think the square with vertices (1,7), (2,9), (3,6) and (4,8) is smaller because its area is 5
* the smallest rectangle has area 8 but I think the rectangle with vertices (1,7), (2,8), (3,5) and (4,6) is smaller because its area is 4
Thank you, Cecilia. Nicely done! I agree those are smaller, and I’ve changed the answers. Note that the digits are up to 8, so your square solution doesn’t work, but there is still a square with an area of 5.