Question 1: What are the coordinates of a point (x, y) after a 180° rotation about the origin?

Answer: (-x, -y)

Question 2: Rotate the point (3, -2) 180° about the origin. Write the new coordinates.

Answer: (-3, 2)

Question 3: Plot the point (4, 5). Rotate it 180° about the origin and describe the location of the new point.

Answer: The new point is at (-4, -5).

Question 4: A triangle has vertices at (1, 2), (4, 2), and (2, 5). Construct the image of this triangle after a 180° rotation about the origin. List the new vertices.

Answer: (-1, -2), (-4, -2), (-2, -5)

Question 5: Explain why rotating a point (x, y) 180° about the origin results in the point (-x, -y).

Answer: Rotation of 180° about the origin changes the sign of both coordinates because it effectively flips the point across both axes, mapping (x, y) to (-x, -y).

Question 6: A square has vertices at (2, 3), (2, 5), (4, 5), and (4, 3). Construct its image after a 180° rotation about the origin. State all new vertices.

Answer: (-2, -3), (-2, -5), (-4, -5), (-4, -3)

Question 7: If a point (x, y) is rotated 180° about the origin to (-x, -y), what is the image of the point (-7, 4)?

Answer: (7, -4)

Question 8: A kite has vertices at (1, 1), (3, 4), (5, 1), and (3, -2). Find the coordinates of the kite after a 180° rotation about the origin.

Answer: (-1, -1), (-3, -4), (-5, -1), (-3, 2)

Question 9: Identify the common mistake students might make when rotating a point about the origin and suggest how to correct it.

Answer: A common mistake is reversing the signs incorrectly or only changing one coordinate. To correct this, students should remember both coordinates change signs simultaneously for a 180° rotation.

Question 10: A real-world scenario: A robot at position (6, -3) rotates 180° about the origin. Where does it end up? How can this movement be useful in navigation?

Answer: The robot moves to (-6, 3). This movement can be useful for retracing a path or performing a turn in navigation systems.

Question 11: Challenge: Prove that performing two 180° rotations about the origin returns a point to its original position.

Answer: Performing two rotations: first (x, y) to (-x, -y), then again to (x, y). Therefore, rotating twice by 180° about the origin returns the point to its original position.