Question 1: Rotate the point (3, 4) about the origin by 180°. What are the coordinates of the new point?

Answer: (-3, -4)

Question 2: A triangle has vertices at (2, 1), (4, 3), and (2, 5). After rotating about the origin by 180°, what are the new coordinates of the vertices?

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

Question 3: Construct a point that is the image of (−2, 6) after a 180° rotation about the point (1, 2).

Answer: (4, -2)

Question 4: Plot the point (−5, 3) on the grid. Rotate it 180° about the origin and mark the new position. What are the coordinates?

Answer: (5, -3)

Question 1: A square has vertices at (1,1), (1,3), (3,3), and (3,1). If the square is rotated 180° about its center at (2,2), what are the new coordinates of each vertex?

Answer: Vertices will be at (3,3), (3,1), (1,1), and (1,3)

Question 2: Explain how rotating a point (x, y) 180° about a point (a, b) transforms its coordinates. Derive the general formula.

Answer: The image is at (2a - x, 2b - y)

Question 3: A triangle with vertices at (−1, 2), (3, 4), and (2, 0) is rotated 180° about the point (1, 2). Find the new coordinates of each vertex.

Answer: (3, 2), (−1, 0), (0, 4)

Question 4: Describe a real-world situation where rotating an object 180° about a point might be used, and explain the steps involved.

Answer: Possible answer: Flipping a sign or object to face the opposite direction around a pivot point, involving identifying the pivot and rotating all points 180°.

Question 1: A robot arm is positioned at (4, 2) and needs to rotate 180° around a fixed point at (2, 2). What will be the robot arm's new position?

Answer: 0, 2

Question 2: A mural painter is rotating a design 180° around a central point to create a mirror image. If a point on the design is at (6, -3), where will it be after rotation?

Answer: (-6, 3)

Question 3: Explain how understanding rotation about a point can help in designing symmetrical objects or patterns in architecture.

Answer: It helps in creating mirror images and symmetry by rotating parts around a point, ensuring balanced and aesthetically pleasing designs.

Question 1: Prove that rotating a point (x, y) 180° about any point (a, b) results in the coordinate (2a - x, 2b - y).

Answer: The transformation is derived from shifting the point to the origin, rotating, then shifting back.

Question 2: Given a parallelogram with vertices at (1,2), (4,2), (5,5), and (2,5), if it is rotated 180° about the point (3,3), find the new coordinates of each vertex.

Answer: (5, 4), (0, 4), (−2, 1), (1, 1)

Question 1: A point at (−3, 5) is rotated 180° about the point (0, 0). What are the coordinates of the image?

Answer: (3, -5)

Question 2: Construct on the grid the image of the point (2, -4) after a 180° rotation about the point (2, 2).

Answer: (2, 8)

Question 1: A student claims that rotating (4, 5) about (2, 2) by 180° results in (−4, −5). Is this correct? If not, what is the correct answer?

Answer: Incorrect. The correct coordinates are (0, -1).

Question 2: A student forgets to shift the point before rotating. Describe the mistake and explain the correct process.

Answer: The mistake is rotating around the origin instead of the specified point. The correct method is to shift the point so the rotation center is at the origin, then rotate, then shift back.