Question

select all the sequences of transformations that could return a figure to its original position. reflect the figure over a line and then reflect back over the same line. translate the figure 1 unit to the right, then 4 units to the left, and then 3 units to the right. reflect the figure over one line and then reflect over a different line. rotate the figure 90° counterclockwise around a point and then 270° counterclockwise around the same point. rotate the figure 90° counterclockwise around a point and then 270° clockwise around the same point.

Explanation:

Step1: Analyze first option

Reflecting a figure over a line and then reflecting back over the same line is the inverse operation. It will return the figure to its original position.

Step2: Analyze second option

Translate 1 unit right, 4 units left, and 3 units right. Net translation: \(1 - 4+3=0\) units. So the figure returns to its original position.

Step3: Analyze third option

Reflecting over one line and then a different line is not guaranteed to return the figure to its original position. It depends on the relationship between the two lines.

Step4: Analyze fourth option

Rotate 90° counter - clockwise and then 270° counter - clockwise around the same point. Total rotation: \(90 + 270=360^{\circ}\), which returns the figure to its original orientation and position.

Step5: Analyze fifth option

Rotate 90° counter - clockwise and then 270° clockwise around the same point. Net rotation: \(90-270=- 180^{\circ}\), which does not return the figure to its original position.

Answer:

Reflect the figure over a line and then reflect back over the same line.
Translate the figure 1 unit to the right, then 4 units to the left, and then 3 units to the right.
Rotate the figure 90° counterclockwise around a point and then 270° counterclockwise around the same point.