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\( 9: 02 \) 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? \( \mathrm{J}^{\prime}(4,4) \) \( \mathrm{J}^{\prime}(-4,4) \) \( \mathrm{J}^{\prime}(4,-4) \)

Ask by Marshall Donnelly. in the United States
Mar 20,2025

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Answer

After a \( 180^{\circ} \) rotation about the origin, the coordinates of vertex \( J \) are \( (4,4) \).

Solution

To perform a \( 180^{\circ} \) rotation about the origin, we use the rule that any point \( (x, y) \) becomes \( (-x, -y) \). Given the vertex \( J(-4,-4) \), we apply the rule as follows: \[ (-4,-4) \rightarrow (-(-4), -(-4)) = (4,4) \] Thus, the coordinates of vertex \( J \) after a \( 180^{\circ} \) rotation are \( \boxed{(4,4)} \).

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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!

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