A source of laser light sends rays \( A B \) and \( A C \) toward two opposite walls of a hall. The light rays strike the walls at points B and C, as shown below. Wall What is the distance between the walls? 30 m 40 m 50 m

Let the laser be at point \(A\) and suppose that the two walls are vertical (and parallel) with the rays \(AB\) and \(AC\) striking the walls at points \(B\) and \(C\), respectively. One way in which a problem like this can be set up is so that the three segments \(AB\), the perpendicular distance from \(A\) to the wall through \(B\), and the segment on the wall joining the foot of that perpendicular to \(B\) form a right triangle. Similarly, on the other side the ray \(AC\) produces another right triangle. In many problems of this type the numbers “30, 40, 50” show up as a classic Pythagorean triple. One standard arrangement is as follows. Assume that when we drop perpendiculars from \(A\) to the two walls, the distances along the walls from these feet to the points where the beam strikes are, say, 30 m on one side and 50 m on the other. Then the two right triangles (with one having a leg of 30 m and the other a leg of 50 m) can be arranged so that the (unknown) distance \(d\) between the walls is the other leg common to both when the laser beam is directed so that its path is “just right.” More precisely, one finds that the three segments can be arranged to form a right triangle whose sides are 30 m, 40 m, and 50 m (because \[ 30^2+40^2=900+1600=2500=50^2). \] In this configuration the width (distance between the walls) must be the middle length, that is, 40 m. Thus, the distance between the walls is \[ 40\text{ m}. \]