Question 1210201
<r>
Your link does not show a diagram....<br>
The AI solution from the other tutor is wrong.  That solution says the ray hits vertex D after the single reflection off side BC.  That will not happen if the length of BC is 2/3.  (Note that, by symmetry, the ray would hit vertex D after the single reflection off side BC if the length of BX were 1/2 instead of 2/3.)<br>
Let Y be the point on CD where the ray hits after reflecting off side BC.<br>
Since the angle of reflection is the same as the angle of incidence, triangles ABX and YCX are similar.  Since BX is 2/3 and CX is 1/3, the ratio of similarity between the two triangles is 2:1.<br>
But then, by that ratio of similarity, CY is half the length of AB, which is a side of the square.  So Y is the midpoint of CD.<br>
From there, we can see by symmetry that the ray will reflect off side CD and continue to vertex B.<br>
Here is a diagram, with the path of the ray in red, starting from vertex A and ending at vertex B.<br>
{{{drawing(500,500,0,5,0,5,
line(1,1,4,1),line(4,1,4,4),line(4,4,1,4),line(1,4,1,1),
red(line(1,1,4,3)),red(line(4,3,2.5,4)),red(line(2.5,4,1,3)),red(line(1,3,4,1)),
locate(.9,.9,A),locate(4.1,.9,B),locate(4.1,4.1,C),locate(.9,4.1,D),
locate(4.1,3,X),locate(2.5,4.2,Y),
locate(2.5,.9,1),locate(4.1,2.2,2/3),locate(4.1,3.7,1/3),locate(3.4,4.4,1/2)
)}}}<br>
From the Pythagorean Theorem with AB=1 and BX=2/3, the length of AX is {{{sqrt(13)/3}}}.<br>
Because of the similarity of triangles ABX and YCX, the length of XY is {{{sqrt(13)/6}}}.<br>
That makes the length of the path of the ray from A to X to Y {{{sqrt(13)/3+sqrt(13)/6=sqrt(13)/2}}}.<br>
And then by symmetry the path from Y to side AD and then on to vertex B is again {{{sqrt(13)/2}}}.<br>
So the total length of the path from vertex A to vertex B, reflecting off sides BC, CD, and DA, is {{{sqrt(13)}}}.<br>
ANSWER: {{{sqrt(13)}}}<br>