Question 1: Calculate the coordinates of point P(3, 4) after a 180° rotation about the origin.

Answer: (-3, -4)

Question 2: Construct the image of triangle ABC with vertices A(1,2), B(4,5), C(2,4) after a 180° rotation about point D(2,2).

Answer: Vertices at A'(-1, 0), B'(-2, 1), C'(0, 0)

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

Answer: Vertices at (3,3), (1,3), (1,1), (3,1)

Question 4: Explain why rotating a point 180° about a fixed point always results in a point directly opposite the original across that point.

Answer: Because a 180° rotation about a point swaps the position of a point with its mirror image across that point, resulting in points being directly opposite each other across the center of rotation.

Question 1: Point P(5, -2) is rotated 180° about point Q(1, 3). Find the coordinates of P' and verify that Q is the midpoint of the segment PP'.

Answer: P'(-3, 8); Q is midpoint of PP' as ( (5 + -3)/2, (-2 + 8)/2 ) = (1, 3)

Question 2: A kite has vertices at (0,0), (4,0), (3,3), (1,3). If it is rotated 180° about (2,1), describe the transformation and the position of the new vertices. Explain the significance of choosing the point (2,1).

Answer: The kite is rotated 180° about (2,1), swapping each vertex with its opposite across this point, resulting in vertices at (4,2), (0,2), (1,-1), (3,-1). The point (2,1) acts as the center of rotation, ensuring symmetry.

Question 3: A rectangle with vertices at (2,2), (6,2), (6,4), (2,4) is rotated 180° about its center. Find the new vertices and describe the transformation.

Answer: Vertices at (2,2), (6,2), (6,4), (2,4); the rectangle maps onto itself because rotation about the center results in the same set of vertices.

Question 1: A drone is flying at coordinates (8, 3). It needs to return to its original position after flying in a straight line. If the drone’s route is rotated 180° about a point at (5, 5), what is its new position? Explain the relevance of rotation in coordinate transformations for navigation.

Answer: The drone’s new position is at (2, 7). Rotation helps in planning symmetrical routes or returning to a starting point by applying geometric transformations.

Question 1: Construct a regular hexagon with side length 4 units. Rotate it 180° about a vertex. Describe the resulting figure and prove that the shape remains a regular hexagon after the rotation.

Answer: The hexagon is rotated around the vertex, mapping each other vertex to a position directly opposite across that point, maintaining side lengths and angles, thus remaining a regular hexagon.

Question 2: Two figures are rotated 180° about the same point but in opposite directions. Explain how their positions relate and analyze whether they are congruent after the rotations.

Answer: Rotations of 180° about the same point in opposite directions produce figures that are congruent and mirror images across the center of rotation, maintaining size and shape.

Question 1: A student claims that rotating a point (x, y) 180° about the origin results in (x, y). Is this correct? Explain your reasoning and correct the mistake if necessary.

Answer: No, because rotating 180° about the origin maps (x, y) to (-x, -y); the original point remains fixed only if (x, y) = (0, 0).