Daily Math Puzzle: 2026-07-10

90 degrees — Let the square's corners be A, B, C, D in clockwise order, and let O be the geometric center of the square. The laser emitter starts at Corner A. At this point, the laser beam points from A to O. As the cart moves along the first side to Corner B, the laser emitter's position changes. When the cart reaches Corner B, the laser beam points from B to O. The question asks for the total angle the laser beam rotates, which means the angle between its initial direction (vector AO) and its final direction (vector BO). Consider the lines segments AO and BO. In a square, the diagonals bisect each other at the center. If you connect two adjacent corners of a square (A and B) to its center (O), the angle formed at the center (angle AOB) is exactly 90 degrees. This is because a square has four equal sides and four right angles, and its diagonals divide it into four congruent right-angled isosceles triangles (if O is the origin, A=(-50,50), B=(50,50), then vector AO=(50,-50) and vector BO=(-50,-50). The dot product AO · BO = (50)(-50) + (-50)(-50) = -2500 + 2500 = 0. A dot product of 0 indicates the vectors are perpendicular). Therefore, the laser beam rotates by an angle of 90 degrees.