A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance the beam will travel along this path?

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We have a laser reflection problem. A laser is positioned at point (3,5) and needs to reach the target at point (7,5). However, the laser beam cannot travel directly. Instead, it must first bounce off the y-axis, then bounce off the x-axis, before finally reaching its destination. Our goal is to find the total distance the laser beam will travel along this specific bouncing path. To solve this reflection problem efficiently, we use the mirror reflection method. This technique involves creating mirror images of our target point across the reflecting surfaces. First, we mirror the target point (7,5) across the y-axis to get (-7,5). Then we mirror this new point across the x-axis to get (-7,-5). The optimal path is then a straight line from the laser to this final mirror image. Now we find where the straight line from (3,5) to (-7,-5) intersects the axes. The slope of this line is negative 10 over negative 10, which equals 1. So our line equation is y equals x plus 2. This line intersects the y-axis at (0,2) and the x-axis at (-2,0). These are our reflection points. Now we calculate the total distance. The laser travels from (3,5) to (0,2), which is 3 root 2. Then from (0,2) to (-2,0), which is 2 root 2. Finally from (-2,0) to (7,5), which is root 106. The total distance is 5 root 2 plus root 106, which equals approximately 17.4 units. The law of reflection states that the angle of incidence equals the angle of reflection. In coordinate geometry, this translates to simple rules: when reflecting across the y-axis, we change the sign of the x-coordinate while keeping y the same. When reflecting across the x-axis, we change the sign of the y-coordinate while keeping x the same. For example, point (2,1) becomes (-2,1) when reflected across the y-axis, and (2,-1) when reflected across the x-axis. The mirror image method provides an elegant solution to this reflection problem. Instead of calculating complex bouncing angles, we create mirror images of our target point. First, we reflect the target (7,5) across the x-axis to get (7,-5). Then we reflect this new point across the y-axis to get (-7,-5). The key insight is that the shortest bouncing path from the laser to the target equals the straight-line distance from the laser to this final mirror image at (-7,-5). Now we calculate the final distance using the distance formula. The distance from (3,5) to (-7,-5) is the square root of (-7 minus 3) squared plus (-5 minus 5) squared. This equals the square root of (-10) squared plus (-10) squared, which is the square root of 100 plus 100, or the square root of 200. Since 200 equals 100 times 2, this simplifies to 10 times the square root of 2. Therefore, the total distance the laser beam travels is 10√2, which is approximately 14.14 units. Let's verify our complete laser path. The laser starts at (3,5), travels to the y-axis at (0,2), reflects and travels to the x-axis at (4,0), then reflects again to reach the target at (7,5). Each reflection follows the law that the angle of incidence equals the angle of reflection. The total distance is indeed 10√2, approximately 14.14 units. The mirror image method has elegantly transformed what could have been a complex reflection problem into a simple straight-line distance calculation, demonstrating the power of geometric insight in problem solving.