document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210201>Question 1210201</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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\r<BR>\n" );
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document.write( "https://artofproblemsolving.com/texer/zqcbfanp </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851581 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13195)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851581\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <r><BR>\n" );
document.write( "Your link does not show a diagram....<br><BR>\n" );
document.write( "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><BR>\n" );
document.write( "Let Y be the point on CD where the ray hits after reflecting off side BC.<br><BR>\n" );
document.write( "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><BR>\n" );
document.write( "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><BR>\n" );
document.write( "From there, we can see by symmetry that the ray will reflect off side CD and continue to vertex B.<br><BR>\n" );
document.write( "Here is a diagram, with the path of the ray in red, starting from vertex A and ending at vertex B.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C0%2C5%2C0%2C5%2C%0D%0Aline%281%2C1%2C4%2C1%29%2Cline%284%2C1%2C4%2C4%29%2Cline%284%2C4%2C1%2C4%29%2Cline%281%2C4%2C1%2C1%29%2C%0D%0Ared%28line%281%2C1%2C4%2C3%29%29%2Cred%28line%284%2C3%2C2.5%2C4%29%29%2Cred%28line%282.5%2C4%2C1%2C3%29%29%2Cred%28line%281%2C3%2C4%2C1%29%29%2C%0D%0Alocate%28.9%2C.9%2CA%29%2Clocate%284.1%2C.9%2CB%29%2Clocate%284.1%2C4.1%2CC%29%2Clocate%28.9%2C4.1%2CD%29%2C%0D%0Alocate%284.1%2C3%2CX%29%2Clocate%282.5%2C4.2%2CY%29%2C%0D%0Alocate%282.5%2C.9%2C1%29%2Clocate%284.1%2C2.2%2C2%2F3%29%2Clocate%284.1%2C3.7%2C1%2F3%29%2Clocate%283.4%2C4.4%2C1%2F2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "From the Pythagorean Theorem with AB=1 and BX=2/3, the length of AX is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2813%29%2F3 ALIGN=MIDDLE  ALT=\"sqrt%2813%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Because of the similarity of triangles ABX and YCX, the length of XY is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2813%29%2F6 ALIGN=MIDDLE  ALT=\"sqrt%2813%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "That makes the length of the path of the ray from A to X to Y <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2813%29%2F3%2Bsqrt%2813%29%2F6=sqrt%2813%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%2813%29%2F3%2Bsqrt%2813%29%2F6=sqrt%2813%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "And then by symmetry the path from Y to side AD and then on to vertex B is again <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2813%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%2813%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "So the total length of the path from vertex A to vertex B, reflecting off sides BC, CD, and DA, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2813%29 ALIGN=MIDDLE  ALT=\"sqrt%2813%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2813%29 ALIGN=MIDDLE  ALT=\"sqrt%2813%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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