Question 1210201
.
Let ABCD be a square with side length 1. A laser is located at vertex A, which fires a laser beam 
at point X on side BC, such that BX = 2/3. The beam reflects off the sides of the square, 
until it ends up at another vertex; at this point, the beam will stop. 
Find the length of the total path of the laser beam. The diagram is linked below

https://artofproblemsolving.com/texer/zqcbfanp
~~~~~~~~~~~~~~~~~~~~~~~~



        Couple of notices before to start.


            (a)    The link is  EMPTY  and does not show/(does not contain)  any picture/diagram.


            (b)    The solution and all calculations in the post by @CPhill are WRONG.


        I came to bring a correct solution.



<pre>
(1)  First beam goes from A to X on BC.

     One leg is 1 unit, other leg is 2/3 of the unit.

     Hence,  AX = {{{sqrt(1^2 + (2/3)^2)}}} = {{{sqrt(1+4/9)}}} = {{{sqrt(13)/3}}}



(2)  Then the beam reflects at X on BC and goes to point Y on DC.

     The angle of incidence at X is equal to the angle of reflection at X.

     Hence, right-angled triangles ABX and XCY are similar.

     The similarity coefficient is  BX/XC = {{{((2/3))/((1/3))}}} = 2.

     Hence, XY is half of AX. i.e. {{{sqrt(13)/6}}}.



(3)  Next, the beam reflects at Y on DC and goes to point Z on AD.

     Again, the angle of incidence at Y is equal to the angle of reflection at Y.

     Hence, right-angled triangles XCY and YDZ are similar.

     The similarity coefficient is  CY/YD = {{{0.5/0.5}}} = 1.

     So, the triangles XCY and YDZ are not only similar - - - they are CONGRUENT.

     It implies that DZ = 1/3  and ZY = XY = {{{sqrt(13)/6}}}.



(4)  Now point Z is symmetric to point X.

     It means that after next reflection at Z, the beam will go directly to vertex B.
     and the journey will be complete.

     The last beam's interval YB is the same long as AZ, i.e. {{{sqrt(13)/3}}}.



(5)  The last step is to find the total length of the journey 

         AX + XY + YZ + ZB = {{{sqrt(13)/3}}} + {{{sqrt(13)/6}}} + {{{sqrt(13)/6}}} + {{{sqrt(13)/3}}} = {{{sqrt(13)}}},  or about  3.60555.


It is the <U>ANSWER</U> to the problem's question.
</pre>

Solved.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The work and the calculations in the post by @CPhill 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;amaze me with their low level.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;His work/technique is at the level of a student, 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;whose scores are between 2 and 3 in the five-score scale.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Closer to &nbsp;2 &nbsp;than to &nbsp;3.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The true &nbsp;AI &nbsp;level should be &nbsp;6 + + +.