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

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

• Well it the shortest path would be path followed by law according to geometrical laws. When we derive the law for geometrical optics we assume light goes from one point to another taking the shortest path, you can check out the proofs if you want . – aryan bansal Aug 27 at 17:31
• Considering that the shortest path is the one taken by light, will reduce the equation I obtained to one variable and make it easier to solve...I'll try it out, Thanks a lot for your help @aryan bansal – Maven Aug 27 at 18:06

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?