Question

1. which of these sequences of transformations would return a shape to its original position? select all that apply

a. translate 3 units up, then 3 units down.
b. reflect over line p, then reflect over line p again.
c. translate 1 unit to the right, then 4 units to the left, then 3 units to the right.
d. rotate 180° counterclockwise around center c, then rotate 180° counterclockwise around c again.
e. dilate with a scale factor of 2 and translate 2 units up.

Explanation:

Step1: Analyze Option a

A translation of 3 units up and then 3 units down is a reverse translation. The net displacement is \( 3 - 3 = 0 \) units vertically, so the shape returns to its original position.

Step2: Analyze Option b

Reflecting a shape over the same line twice. The first reflection flips the shape, the second reflection flips it back, so it returns to the original position.

Step3: Analyze Option c

Calculate the net horizontal translation: \( 1 - 4 + 3 = 0 \). So the total horizontal displacement is 0, and the shape returns to its original position.

Step4: Analyze Option d

Rotating \( 180^\circ \) counterclockwise twice around the same center. A \( 180^\circ \) rotation twice is equivalent to a \( 360^\circ \) rotation, which brings the shape back to its original position.

Step5: Analyze Option e

Dilation with a scale factor of 2 changes the size of the shape, and a translation of 2 units up. Since the size is changed, the shape does not return to its original position.

Answer:

a. Translate 3 units up, then 3 units down.
b. Reflect over line \( p \), then reflect over line \( p \) again.
c. Translate 1 unit to the right, then 4 units to the left, then 3 units to the right.
d. Rotate \( 180^\circ \) counterclockwise around center \( C \), then rotate \( 180^\circ \) counterclockwise around \( C \) again.