What is the image of the point \( (5,-3) \) after a rotation of \( 90^{\circ} \) counterclockwise about the origin?

To find the image of the point \( (5, -3) \) after a rotation of \( 90^{\circ} \) counterclockwise about the origin, you can use the rotation formula. The general rule for a \( 90^{\circ} \) counterclockwise rotation is to transform the point \( (x, y) \) to \( (-y, x) \). Applying this rule: - The original point is \( (5, -3) \). - The new coordinates after rotation will be \( (-(-3), 5) = (3, 5) \). Hence, the image of the point \( (5, -3) \) after a \( 90^{\circ} \) counterclockwise rotation about the origin is \( (3, 5) \). --- Did you know that the concept of rotation in mathematics dates back to ancient civilizations? The Greeks, particularly, studied geometric constructions and transformations, laying the groundwork for what we now call geometry. Rotations were not only vital in mathematics but also crucial in navigation and astronomy, influencing how ancient civilizations mapped the stars! In real-world applications, understanding rotations can be incredibly helpful! For instance, in graphic design and animation, rotating objects around a point is essential for creating dynamic visuals. Similarly, in robotics, knowing how to manipulate coordinates through rotation allows robots to navigate and interact effectively with their environments. Embrace rotations in your design projects or coding; the creative possibilities are endless!