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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 $\overline{BC},$ such that $BX = \frac{1}{2}.$  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.

 Mar 27, 2024
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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 $\overline{BC},$ such that $BX = \frac{1}{2}.$  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 beam starts at corner A and bounces off the middle of side BC.   

The next place the reflected beam should strike would be corner D. 

 

The beam makes a right triangle ABX with AX being the hypotenuse. 

 

Using Pythagoras Theorem,                                   AX2  =  12 + 0.52  

                                                                               AX2  =  1.5  

                                                                                 AX  =  1.118   

The reflected beam would be the same length.  

 

So, totalling AX + XD,

the total length of the beam is                  1.118 + 1.118  =  2.236    

.

 Mar 27, 2024

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