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.

Akhain1 Mar 27, 2024

#1**+1 **

*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, AX^{2} = 1^{2} + 0**.**5^{2}

AX^{2} = 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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Bosco Mar 27, 2024