 # Creating Rectangles 2

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

What are the properties of a rectangle?
Which lattice points can be used to create the rectangle? Which lattice points cannot be used?

Because squares are rectangles, we can consider four situations for largest/smallest area, as listed below. For each of these there are several translations/reflections of it.

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

## Pythagorean Inequality

Directions: Using the digits 1 through 6 at most one time each, fill in the …

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