24. Line AB with \( \mathrm{A}(-5 ; 9) \) and \( \mathrm{B}(1 ; 2) \) is reflected in the line \( y=x \). What are the co-ordinates of \( \mathrm{A}^{\prime} \) and \( \mathrm{B}^{\prime} \) ? A \( \quad \mathrm{A}^{\prime}(9 ; 5) \) and \( \mathrm{B}^{\prime}(2 ;-1) \) B \( \quad \mathrm{A}^{\prime}(-5 ; 9) \) and \( \mathrm{B}^{\prime}(2 ; 1) \) C \( \quad \mathrm{A}^{\prime}(5 ;-9) \) and \( \mathrm{B}^{\prime}(-1 ;-2) \) D \( \quad \mathrm{A}^{\prime}(9 ;-5) \) and \( \mathrm{B}^{\prime}(2 ; 1) \)

To find the coordinates of the reflected points \( \mathrm{A}^{\prime} \) and \( \mathrm{B}^{\prime} \) through the line \( y=x \), we simply switch the \( x \) and \( y \) coordinates of each point. For point \( \mathrm{A}(-5, 9) \), reflecting it gives \( \mathrm{A}^{\prime}(9, -5) \). For point \( \mathrm{B}(1, 2) \), it reflects to \( \mathrm{B}^{\prime}(2, 1) \). Thus, the answer is D: \( \mathrm{A}^{\prime}(9 ; -5) \) and \( \mathrm{B}^{\prime}(2 ; 1) \). Reflecting points in coordinate geometry is not just useful in math class; it's a phenomenon used in computer graphics and even in architectural design! So when you’re doodling in your notebook, remember there’s a whole world of applications behind those reflections. If you’re delving into reflections and transformations, consider digging into topics like linear algebra or geometry. Both offer fantastic insights and can deepen your understanding of how objects interact in space—plus, they’re fundamental in fields like robotics and computer vision!