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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 \(BX = \frac{3}{4}\). 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.

 Jul 14, 2020
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I would suggest drawing this on a sheet of graph paper but make the distance from A to B to be 12, not 1.

This will make finding the points of reflection, and the distances, much easier.

However, the last step will be to divide the answer by 12.

 

Place A at the origin:  A = (0, 0)     B = (12, 0)     C = (12, 12)     D = (0, 12)

 

Also, since there will be many points of reflecttion, rename point X as point X1.

 

Since X1B = three-fourths of AB,  X1 = (12, 9).

 

The point X2 will be on side CD.  

The "angle going in" will equal the "angle coming out", so CX2 / CX1  =  AB / X1B.

This makes  X2 = (8, 12).

 

Point X3 will be on side DA   --->   DX3 / DX2  =  CX1 / CX2   --->   X3 = (0, 6).

 

Point X4 will be on side AB   --->   X4 = (8, 0).

 

Point X5 will be on side BC   --->   X5 = (12, 3).

 

Point X6 will be point D   --->   X6 = (0, 12).

 

Now, use the distance formula to find all the distances:

AX1   =  sqrt( (12 - 0)2 + (9 - 0)2 )  =  15

X1X2  = sqrt( (8 - 12)2 + (12 - 9))  =  5

X2X3  =  ...

X3X4  =  ...

X4X5  =  ...

X5D   =  ...

 

Add these distances together and then divide by 12.

 Jul 14, 2020

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