 # Creating Rectangles 2

Directions: Using only the whole numbers 1 through 8 (without repeating any number), 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 can not be used?

There are 4 different sized rectangles. For each of these there are several translations/reflections of it. This list may or may not be exhaustive but it ‘feels’ complete.

The Big Rectangle: area 16, perimeter 12sqrt(2)
(1,4)(3,2)(7,6)(5,8)
(1,6)(3,8)(7,4)(5,2)

The Big Square: area 20, perimeter 8sqrt(5)
(1,6)(5,8)(7,4)(3,2)
(1,4)(3,8)(7,6)(5,2)

The Small Rectangle: area 8, perimeter 10sqrt(2)
(1,4)(5,8)(6,7)(2,3)
(1,7)(2,8)(6,4)(5,3)

The Small Square: area 17, perimeter 4sqrt(17)
(1,4)(2,8)(6,7)(5,3)
(2,3)(6,4)(5,8)(1,7)

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: Use the whole numbers 1 through 6, at most one time each, to find …