Consider the laser beam in The figure below, in which L = 4.2 m and H = 2.1 m. At what angle phi should the b?

I'm going to call the height of the box l Then the length of the box = 2l. The distance between the point where the light reflects from the top surface to the right side will be called p. Then the point of reflection to the left side has the distance be 2l-p. That's because the L is twice H.

So now you have a proportion ready to setup. A right-angle triangle has sides with lengths x, y, and r. The side y is opposite phi.

The triangle on the right side of the box has the same proportions as the triangle formed on the left side of box. that then means that side y divided by side x is the same for both triangles.

1/2 l = l

____ ____

p 2l-p

Solving this, you get p = 2l/3. Now, to calculate phi. The triangle formed by the reflection to the right, using x,y, and r notation has:

y = 1/2 l

x = p but p also = 2l/3 from above

so y/x = tan (pHi) = 1/2 l divided by 2l/3 = 3/4.

Take atan (3/4) and you get phi.