Question

which sequence of rigid motions maps one figure onto the congruent figure? select all that apply. 180 degree rotation around the origin reflection across the y - axis followed by a 90 degree clockwise rotation around the origin 90 degree clockwise rotation around the origin followed by a reflection across the y - axis reflection across the x - axis followed by a 90 degree counterclockwise rotation around the origin 90 degree counterclockwise rotation around the origin followed by a reflection across the x - axis

Explanation:

Step1: Recall transformation rules

For a 180 - degree rotation around the origin, the rule is $(x,y)\to(-x,-y)$. For a reflection across the $y$-axis, the rule is $(x,y)\to(-x,y)$. For a 90 - degree clockwise rotation around the origin, the rule is $(x,y)\to(y, - x)$. For a reflection across the $x$-axis, the rule is $(x,y)\to(x,-y)$. For a 90 - degree counter - clockwise rotation around the origin, the rule is $(x,y)\to(-y,x)$.

Step2: Analyze 180 - degree rotation

Applying a 180 - degree rotation around the origin to all points of one figure will map it onto the congruent figure.

Step3: Analyze reflection + 90 - degree clockwise rotation

If we first reflect across the $y$-axis and then do a 90 - degree clockwise rotation, the combined transformation will not map one figure onto the other.

Step4: Analyze 90 - degree clockwise rotation+ reflection

If we first do a 90 - degree clockwise rotation and then reflect across the $y$-axis, the combined transformation will not map one figure onto the other.

Step5: Analyze reflection across $x$-axis + 90 - degree counter - clockwise rotation

If we first reflect across the $x$-axis and then do a 90 - degree counter - clockwise rotation, the combined transformation will not map one figure onto the other.

Step6: Analyze 90 - degree counter - clockwise rotation+ reflection

If we first do a 90 - degree counter - clockwise rotation and then reflect across the $x$-axis, the combined transformation will not map one figure onto the other.

Answer:

180 degree rotation around the origin