A laser is shot from vertex A of square ABCD of side length 1, towards point P on BC so that BP = 3/4. The laser reflects off the sides of the square, until it hits another vertex, at which point it stops. What is the length of the path the laser takes?
Help?? Won't the laser bounce around until infinity?
No, the laser will not bounce around indefinitely. It will eventually hit another vertex of the square and stop. To find the length of the path the laser takes, let's break down the problem step by step.
1. Start by visualizing the setup. Draw a square ABCD with side length 1. Place point P on BC such that BP = 3/4. This means that BP is shorter than BC.
2. Since the laser starts from vertex A, it will first hit side BC at point P. To determine the angle of incidence, we need to find the angle between the laser and the side BC. We can do this by using the tangent function.
- The tangent of the angle of incidence is equal to the opposite side (PB) divided by the adjacent side (PA).
- PB = 3/4, and PA = 1, since it's the length of one side of the square.
- So the tangent of the angle of incidence is PB/PA = (3/4) / 1 = 3/4.
3. Use the inverse tangent function (arctan) to find the angle of incidence. In this case, the angle of incidence is arctan(3/4) = 36.87 degrees.
4. Now, the laser will reflect off the side BC and change direction. The angle of reflection will be equal to the angle of incidence. So the angle of reflection is also 36.87 degrees.
5. The laser will now continue along the reflected path towards another side of the square. Since we know the length of BC, we can calculate the distance the laser will travel along BC.
- BP = 3/4, and BC = 1 (side length of the square).
- The laser travels a distance of BC - BP = 1 - 3/4 = 1/4 along BC.
6. The laser will then hit another vertex of the square. Since it reflects off the vertices, it will travel a distance equal to the diagonal of the square, which is √2.
7. Finally, the laser stops at the new vertex. We have accounted for all the segments it travels, so we can add up the distances to find the total path length.
- Path length = (distance along BC) + (distance along diagonal) = 1/4 + √2.
Therefore, the length of the path the laser takes is 1/4 + √2.