Question

1. select all the sequences of transformations that could return a figure to its original position.

a. rotate the figure around a point 90° clockwise and then rotate it another 90° clockwise around the same point.
b. reflect the figure over one line and then reflect over a different line.
c. translate the figure 3 units down, then 5 units up, and then 2 units down.
d. reflect the figure over one line and then reflect over the same line.
e. rotate the figure 90° counterclockwise around a point and then 270° counterclockwise around the same point.
f. rotate a figure 180° counterclockwise, then reflect it over a vertical line.

Explanation:

• Option A: Rotating a figure 90° clockwise and then another 90° clockwise around the same point results in a total rotation of 180°, not back to the original position. So A is incorrect.
• Option B: Reflecting over one line and then a different line generally does not return the figure to its original position (unless the lines are perpendicular and the figure has specific symmetry, which is not assumed here). So B is incorrect.
• Option C: Translating 3 units down, then 5 units up (net 2 units up), then 2 units down results in a net translation of 0 (3 down + 5 up = 2 up; 2 up + 2 down = 0). So the figure returns to its original position. C is correct.
• Option D: Reflecting over the same line twice is equivalent to the identity transformation (the figure returns to its original position). D is correct.
• Option E: Rotating 90° counterclockwise and then 270° counterclockwise around the same point: 90° + 270° = 360°, which is a full rotation, so the figure returns to its original position. E is correct.
• Option F: Rotating 180° counterclockwise and then reflecting over a vertical line does not generally return the figure to its original position (the rotation and reflection are not inverse operations in this case). So F is incorrect.

Answer:

C. Translate the figure 3 units down, then 5 units up, and then 2 units down.
D. Reflect the figure over one line and then reflect over the same line.
E. Rotate the figure 90° counterclockwise around a point and then 270° counterclockwise around the same point.