Question

select the sequence of transformations that will carry rectangle a onto rectangle a.
a rotate 90° clockwise, reflect over y-axis, then translate 5 units down
b rotate 90° clockwise, reflect over y-axis, then reflect over line y = 2
c rotate 90° clockwise, reflect over x-axis, then reflect over line x = 5
d rotate 180° clockwise, reflect over x-axis, then reflect over line x = 5

Explanation:

Step1: Analyze rotation

A \(90^{\circ}\) clock - wise rotation changes the orientation of the rectangle. A \(180^{\circ}\) rotation would not lead to the correct orientation as per the figure.

Step2: Analyze reflection

Reflecting over the \(x\) - axis (transformation \((x,y)\to(x, - y)\)) and then reflecting over the line \(x = 5\) (transformation \((x,y)\to(10 - x,y)\)) after a \(90^{\circ}\) clock - wise rotation (transformation \((x,y)\to(y,-x)\)) will map rectangle \(A\) to \(A'\).

• For option A: Translating 5 units down after rotation and reflection over \(y\) - axis will not give the correct position.
• For option B: Reflecting over \(y = 2\) after rotation and reflection over \(y\) - axis will not give the correct position.
• For option D: \(180^{\circ}\) rotation is incorrect as per the orientation.

Answer:

C. rotate \(90^{\circ}\) clockwise, reflect over \(x\) - axis, then reflect over line \(x = 5\)