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.

envoy Jul 14, 2020

#1**0 **

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 X_{1}.

Since X_{1}B = three-fourths of AB, X_{1} = (12, 9).

The point X_{2} will be on side CD.

The "angle going in" will equal the "angle coming out", so CX_{2} / CX_{1} = AB / X_{1}B.

This makes X_{2} = (8, 12).

Point X_{3} will be on side DA ---> DX_{3} / DX_{2} = CX_{1} / CX_{2} ---> X_{3} = (0, 6).

Point X_{4} will be on side AB ---> X_{4} = (8, 0).

Point X_{5} will be on side BC ---> X_{5} = (12, 3).

Point X_{6} will be point D ---> X_{6} = (0, 12).

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

AX_{1} = sqrt( (12 - 0)^{2} + (9 - 0)^{2} ) = 15

X_{1}X_{2} = sqrt( (8 - 12)^{2} + (12 - 9)^{2 }) = 5

X_{2}X_{3} = ...

X_{3}X_{4} = ...

X_{4}X_{5} = ...

X_{5}D = ...

Add these distances together and then divide by 12.

geno3141 Jul 14, 2020