+1246 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.
+1250
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
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