Problem D

From Square to Hexagon

A square piece of paper, \(PQRS\), has side length \(40\) cm. The page is grey on one side and white on the other side. Point \(M\) is the midpoint of side \(PQ\) and point \(N\) is the midpoint of side \(PS\).

The paper is folded along \(MN\) so that \(P\) touches the paper at the point \(P'\).

Point \(T\) lies on \(QR\) and point \(U\) lies on \(SR\) such that \(TU\) is parallel to \(MN\), and when the paper is folded along \(TU\), the point \(R\) touches the paper at the point \(R'\) on \(MN\).

What is the area of hexagon \(NMQTUS\)?

Here are some known properties of the diagonals of a square that may be useful:

the diagonals are equal in length; and

the diagonals right bisect each other; and

the diagonals bisect the corner angles.