Question

i can identify the sequence when a rotation, reflection, and/or translation is applied. (8.g.a.1) 1. here are three pairs of figures. a. which transformation takes figure a to figure b in pair 1: a translation, rotation, or reflection? b. which transformation takes figure a to figure b in pair 2: a translation, rotation, or reflection? c. which transformation takes figure a to figure b in pair 3: a translation, rotation, or reflection? 2. all of these sequences of transformations would return a shape to its original position except? 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 120° counterclockwise around center c, then rotate 220° counterclockwise around c again. explain your reasoning.

Explanation:

Step1: Analyze translations

Translation is a rigid - motion that slides a figure. Translating a figure and then translating it back in the opposite direction will return it to its original position. But in isolation, a single translation does not undo itself to return to the original position without a reverse - translation.

Step2: Analyze reflections

Reflecting a figure over a line and then reflecting it over the same line again returns the figure to its original position. For example, if we reflect a point across a line, and then reflect that new point across the same line, we get back to the original point.

Step3: Analyze rotations

A rotation of 360° around a center returns a figure to its original position. Also, if we have a rotation of an angle $\theta$ followed by a rotation of $360^{\circ}-\theta$ around the same center, the figure will return to its original position.

For question 1:

• Pair 1: The transformation from Figure A to Figure B is a translation and rotation.
• Pair 2: The transformation from Figure A to Figure B is a translation, rotation, and reflection.
• Pair 3: The transformation from Figure A to Figure B is a translation and rotation.

For question 2:

• Option A: Translating 3 units up and then 3 units down is a pair of opposite translations that will return the shape to its original position.
• Option B: Reflecting over line $p$ and then reflecting over line $p$ again will return the shape to its original position.
• Option C: Translating 1 unit to the right, then 4 units to the left, then 3 units to the right is equivalent to $1 - 4+3=0$ net translation. But this is a combination of multiple translations. A single translation does not return the shape to its original position without a reverse - translation.
• Option D: Rotating 120° counter - clockwise and then 220° counter - clockwise around the same center gives a total rotation of $120 + 220=340^{\circ}

eq360^{\circ}$, so it does not return the shape to its original position.

Answer:

1.

• Pair 1: Translation, Rotation
• Pair 2: Translation, Rotation, Reflection
• Pair 3: Translation, Rotation

1. C