Question

select all of the transformations that could have been performed to map abcd onto abcd. a reflection across the line y = x a reflection across the line y = -x a rotation of 180 degrees clockwise about the origin a reflection across the x-axis, and then a reflection across the y-axis a rotation of 270 degrees counterclockwise about the origin, and then a reflection across the x-axis

Explanation:

Step1: Analyze 180° rotation

A 180° clockwise rotation about the origin changes a point \((x,y)\) to \((-x,-y)\). This can map a figure to a position like \(A'B'C'D'\) if the original and image have opposite coordinates, which is consistent with the square's transformation here.

Step2: Analyze x - then y - reflection

Reflecting over the \(x\)-axis changes \((x,y)\) to \((x,-y)\), then reflecting over the \(y\)-axis changes \((x,-y)\) to \((-x,-y)\), same as 180° rotation. So this transformation works.

Step3: Analyze other options

• Reflection over \(y = x\) swaps \(x\) and \(y\), which doesn't match the square's position.
• Reflection over \(y=-x\) changes \((x,y)\) to \((-y,-x)\), not matching.
• Rotation 270° counter - clockwise then \(x\)-reflection: 270° counter - clockwise changes \((x,y)\) to \((y,-x)\), then \(x\)-reflection changes to \((y,x)\), not matching.

Answer:

• a rotation of 180 degrees clockwise about the origin
• a reflection across the x - axis, and then a reflection across the y - axis