2022 AMC 8

Solution 1

The midpoints of the four sides of every rectangle are the vertices of a rhombus whose area is half the area of the rectangle: Note that the diagonals of the rhombus have the same lengths as the sides of the rectangle.

Let and Note that and are the vertices of a rhombus whose diagonals have lengths and It follows that the dimensions of the rectangle are and so the area of the rectangle is

Solution 2

If a rectangle has area then the area of the quadrilateral formed by its midpoints is

Define points and as Solution 1 does. Since and are the midpoints of the rectangle, the rectangle's area is Now, note that is a parallelogram since and As the parallelogram's height from to is and its area is Therefore, the area of the rectangle is