\( 9: 02 \) All changes saved 11. On the coordinate plane, \( \triangle \mathrm{HIJ} \) has vertices located at \( \mathrm{H}(-6,-2), \mathrm{I}(-9,-5) \), and \( \mathrm{J}(-4,-4) \). What are the coordinates of vertex J after a \( 180^{\circ} \) rotation about the origin? \( \mathrm{J}^{\prime}(4,4) \) \( \mathrm{J}^{\prime}(-4,4) \) \( \mathrm{J}^{\prime}(4,-4) \)

To determine the coordinates of vertex J after a \( 180^{\circ} \) rotation about the origin, we can use the rule that a point \((x, y)\) transforms to \((-x, -y)\). For vertex J at \((-4, -4)\), applying the rotation gives us \(J' = (4, 4)\). So, the correct answer is \( \mathrm{J}^{\prime}(4, 4) \). Now, if you're ever unsure about transformations, remember to visualize your rotation on graph paper! This not only helps in understanding but makes the whole process feel like a fun little dance across the plane! Speaking of geometry, this transformation technique isn't just for triangles! Many real-world applications - like robotics and game designing - use similar transformations to manipulate shapes and objects. Rotate, reflect, and scale to create stunning designs!