Question 1: A triangle has vertices at A(2, 3), B(4, 5), and C(3, 7). If the triangle is rotated 180° about point P(3, 4), what are the coordinates of the image of point A? Show your working.

Answer: A' (4, 5)

Question 2: A student claims that rotating a point (x, y) 180° about the origin results in the point (-x, -y). Is this statement always true? Explain why or why not.

Answer: Yes, when rotating about the origin, the point (x, y) maps to (-x, -y). This is because rotation 180° about the origin changes both coordinates sign.

Question 3: Construct the image of point D(5, 2) after a 180° rotation about point Q(2, 2). Plot both points on the grid.

Answer: D' (−1, 2)

Question 4: A common misconception is that rotation 180° about any point simply swaps the coordinates. Identify and correct this misconception.

Answer: This is incorrect because rotation 180° about a specific point involves translating the point to the origin, rotating, then translating back, not just swapping coordinates.

Question 5: Calculate the image of point E(1, 4) when rotated 180° about point R(4, 1). Show the steps.

Answer: E' (7, -2)

Question 6: In a diagram, a student rotates a triangle 180° about a point and gets an incorrect image. The student claims the image should be the same as the original. What mistake did the student make?

Answer: The student confused rotation with reflection or did not rotate properly about the specified point, resulting in an incorrect image.

Question 7: Given points F(6, 3) and G(2, 7), find the image of G after a 180° rotation about F. Show your calculation.

Answer: G' (-2, -3)

Question 8: A diagram shows a point at (3, 5). The student claims rotating 180° about (3, 5) leaves the point unchanged. Is this correct? Explain.

Answer: Yes, rotating any point about itself results in the same point because the rotation is centered at that point.

Question 9: Identify and explain a common mistake students make when performing a rotation of 180° about a point that is not the origin.

Answer: Students often forget to translate the point so that the axis of rotation is at the origin, leading to incorrect results.

Question 10: A student attempts to rotate a shape 180° about point S(0, 0), but the image appears to be incorrect. The student says they followed the steps correctly. What should you check?

Answer: Check if the student correctly applied the rotation formula or if they mistakenly swapped coordinates without considering the rotation rules.

Question 11: Challenge: Prove that rotating a point (x, y) 180° about any point P(a, b) results in the image (2a - x, 2b - y).

Answer: Using translation, rotation, and reverse translation, the image is (2a - x, 2b - y).

Question 12: Explain why the image of a point after a 180° rotation about a different point always lies directly opposite to the original point with respect to the rotation point.

Answer: Because rotation 180° about a point creates an image directly opposite the original point with respect to that center, maintaining equal distance on the opposite side.