Final Answer:
The final path of the laser after reflecting off all three mirrors and the y-axis is (8, -1).
Step-by-step explanation:
The laser is initially placed at (2,-1) and is aimed at Mirror 1. Mirror 1 is placed at (5,3) and will reflect the laser at a 90° angle towards Mirror 2. Mirror 2 is placed at (8,0) and will also reflect the laser at a 90° angle towards Mirror 3. Mirror 3 is placed at (8,-4) and will reflect the laser towards the y-axis, which will then reflect it back towards the x-axis. This final reflection will result in the laser's path ending at (8,-1).
To calculate this final path, we can use the formula for the reflection of a line: y = mx + b. In this case, the slope (m) of the laser's path is -1/3, as it is reflected off Mirror 1 and Mirror 2 at 90° angles. The y-intercept (b) can be found by plugging in the coordinates of the initial point (2,-1) into the equation. This gives us b = -1 - (-1/3)(2) = -5/3.
Therefore, the equation for the laser's path after reflecting off Mirror 1 and Mirror 2 is y = (-1/3)x - (5/3). To find the point of intersection between this line and Mirror 3 at (8,-4), we can set the equations equal to each other: (-1/3)x - (5/3) = -4. Solving for x, we get x = 8.
Plugging this value of x into the equation for the laser's path, we get y = (-1/3)(8) - (5/3) = -1. This confirms that the laser's path will end at (8,-1).
Next, we need to consider the reflection off the y-axis. Since the y-axis is a vertical line with no slope, the equation for its reflection can be simplified to x = -x. Plugging in the x-value of 8, we get x = -8, which means the final path of the laser will also reflect off the x-axis at (8,-1).
In conclusion, the final path of the laser after reflecting off all three mirrors and the y-axis is (8,-1). This can be confirmed by plotting the points on a coordinate plane and observing the path of the laser as it reflects off each surface. The use of mathematical equations and calculations has allowed us to accurately determine the final answer and understand the path of the laser in this scenario.