document.write( "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\r
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Algebra.Com's Answer #851584 by ikleyn(52776) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Let ABCD be a square with side length 1. A laser is located at vertex A, which fires a laser beam
\n" ); document.write( "at point X on side BC, such that BX = 2/3. The beam reflects off the sides of the square,
\n" ); document.write( "until it ends up at another vertex; at this point, the beam will stop.
\n" ); document.write( "Find the length of the total path of the laser beam. The diagram is linked below\r
\n" ); document.write( "\n" ); document.write( "https://artofproblemsolving.com/texer/zqcbfanp
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\n" ); document.write( "\n" ); document.write( "        Couple of notices before to start.\r
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\n" ); document.write( "\n" ); document.write( "            (a)   The link is EMPTY and does not show/(does not contain) any picture/diagram.\r
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\n" ); document.write( "\n" ); document.write( "            (b)   The solution and all calculations in the post by @CPhill are WRONG.\r
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\n" ); document.write( "\n" ); document.write( "        I came to bring a correct solution.\r
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document.write( "(1)  First beam goes from A to X on BC.\r\n" );
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document.write( "     One leg is 1 unit, other leg is 2/3 of the unit.\r\n" );
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document.write( "     Hence,  AX =  =  = \r\n" );
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document.write( "(2)  Then the beam reflects at X on BC and goes to point Y on DC.\r\n" );
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document.write( "     The angle of incidence at X is equal to the angle of reflection at X.\r\n" );
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document.write( "     Hence, right-angled triangles ABX and XCY are similar.\r\n" );
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document.write( "     The similarity coefficient is  BX/XC =  = 2.\r\n" );
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document.write( "     Hence, XY is half of AX. i.e. .\r\n" );
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document.write( "(3)  Next, the beam reflects at Y on DC and goes to point Z on AD.\r\n" );
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document.write( "     Again, the angle of incidence at Y is equal to the angle of reflection at Y.\r\n" );
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document.write( "     Hence, right-angled triangles XCY and YDZ are similar.\r\n" );
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document.write( "     The similarity coefficient is  CY/YD =  = 1.\r\n" );
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document.write( "     So, the triangles XCY and YDZ are not only similar - - - they are CONGRUENT.\r\n" );
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document.write( "     It implies that DZ = 1/3  and ZY = XY = .\r\n" );
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document.write( "(4)  Now point Z is symmetric to point X.\r\n" );
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document.write( "     It means that after next reflection at Z, the beam will go directly to vertex B.\r\n" );
document.write( "     and the journey will be complete.\r\n" );
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document.write( "     The last beam's interval YB is the same long as AZ, i.e. .\r\n" );
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document.write( "(5)  The last step is to find the total length of the journey \r\n" );
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document.write( "         AX + XY + YZ + ZB =  +  +  +  = ,  or about  3.60555.\r\n" );
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document.write( "It is the ANSWER to the problem's question.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "        The work and the calculations in the post by @CPhill \r
\n" ); document.write( "\n" ); document.write( "                    amaze me with their low level.\r
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\n" ); document.write( "\n" ); document.write( "                His work/technique is at the level of a student,
\n" ); document.write( "        whose scores are between 2 and 3 in the five-score scale.\r
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\n" ); document.write( "\n" ); document.write( "                        Closer to 2 than to 3.\r
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\n" ); document.write( "\n" ); document.write( "                The true AI level should be 6 + + +.\r
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