I was doing the following problem and my intuition was as follows. If point P touched Point R, the crease would be no distance from Q. When point P touches S, then the crease would be 20 cm away from point Q. So when point P touches point M, then the crease should be 10 cm away from point Q because M is the midpoint. But when I actually folded it out, I found the answer is actually 5cm from point Q.
How could I work this out without folding paper in an exam, what is the intuition behind it?
The crease is the perpendicular bisector of $PM$, which is $AB$.
In the red right triangle, the width is twice the height, and the width is $30$.
So you can easily find the location of $B$.