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avatar+1071 

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.

 Dec 28, 2023
 #1
avatar+129185 
+1

Let A =  (0,1)

Let B = (1,1)

Let X = ( 1, 1/2)

Let D = (0,0)

The beam will reflect from  X to D

 

The path length 2 * AX  = 2 sqrt [ ( 0 -1)^2 + (1 -1/2)^2 ]  = 2 sqrt [ 1 + 1/4] =  2 sqrt (5) / 2  = sqrt 5

 

cool cool cool

 Dec 28, 2023

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