+223 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.
+23252 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)2 ) = 5
X2X3 = ...
X3X4 = ...
X4X5 = ...
X5D = ...
Add these distances together and then divide by 12.
