Question

which of these sequences of transformations will always return a figure to its original position? reflect over line m, then reflect over line m again. rotate 100° counterclockwise around a different point b. then rotate 80° counterclockwise around point b. rotate 90° counterclockwise around a different point b. then rotate 90° counterclockwise around point b again. translate 2 units up, then translate 2 units up again.

Explanation:

Step1: Analyze reflection

Reflecting a figure over a line twice is equivalent to the identity transformation. If we reflect a point \(P\) over a line \(m\) to get \(P'\), and then reflect \(P'\) over the same line \(m\) again, we get back to the original point \(P\).

Step2: Analyze rotation

Rotating \(100^{\circ}\) counter - clockwise and then \(80^{\circ}\) counter - clockwise around the same point \(B\) gives a total rotation of \(100^{\circ}+ 80^{\circ}=180^{\circ}\), not the identity transformation. Rotating \(90^{\circ}\) counter - clockwise twice around a point \(B\) gives a total rotation of \(180^{\circ}\), not the identity transformation.

Step3: Analyze translation

Translating 2 units up twice moves the figure 4 units up, not back to its original position.

Answer:

Reflect over line \(m\), then reflect over line \(m\) again.