+0

# Laser

0
6
1
+1121

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.

Mar 27, 2024

#1
+810
+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,                                   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

.

Mar 27, 2024