Consider a Quadrilateral PQRS as given in the diagram with paths AB,BC and CD as shown. Then what is the minimum value of $(AB+BC+CD)^2 -20$?

Question

I tried using coordinate geometry to find the distance AB,BC and CD and then finding their minimum by partially differentiating the equations. This yielded a rather complicated equation which I could not solve.

I'm sure there must be a more direct and simpler approach to this question, but I am simply hitting dead ends.

Any hints on how to solve the question?

Thanks a lot in advance!

Regards

Answer 1

Your given problem is to compute the shortest path from $A$ to $D$, given the requirement that the path must touch the lines $QR$ and $PS$.

Consider a slightly different problem: let $A'$ be the reflection of $A$ about the line $QR$, and let $D'$ be the reflection of $D$ about the line $PS$. What is the shortest path from $A'$ to $D'$, given the requirement that the path must touch the lines $QR$ and $PS$? Can you solve this problem? Are the two problems related in any way?