Shortest distance around a pyramid

Transcript:

The diagram shows a square based pyramid with base PQRS and vertex O. All the edges are length 20 meters. Find the shortest distance, in meters, along the outer surface of the pyramid from P to the midpoint of OR.

The only way I have been able to solve this question is using a computer and the paper is non calculator, so there must be a faster, better solution.

My solution is creating a point X on OQ and labelling the midpoint of OR as M, and setting $\theta = OPQ$. I then calculated $PX + PM$ in terms of $\theta$ and found the minimum point of this function, in order to find the shortest possible distance. Finding the minimum point would be nowhere near possible under time constraint without a computer.

I have included the correct answer, so it is the working I am looking for.

Answer:

D $10\sqrt7$

Solution 1:

Hint:

The distance has to be a straight line along the net of the pyramid. There are two options:

The first line has lenght $10\sqrt{7+2\sqrt{3}}$, and the second one has $10\sqrt{7}$, which is the smaller one.

Solution 2:

Draw the net of the pyramid and use the cosine rule.

Solution 3:

Hint: Draw the net diagram of the square pyramid and the you will find two points that are mid-point of $OR$. As you know that, minimum distance between any two points is the straight line joining them, just connect $P$ and midpoint of $OR$ by a straight line. Then just find the length of this line and exclude the length of this line that goes out of the net diagram. This length of line will be minimum length. Why?

Note: You can choose any one of the two midpoint because since the diagram is symmetric, the two lengths will be the same.