Question

1. here are three pairs of figures.

pair 1
pair 2
pair 3
a. which transformation takes figure a to figure d 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?

1. identify the sequence when a rotation, reflection, and/or translation is applied. (b.g.a.1)

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 q, 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.

Explanation:

Step1: Analyze translation - only

Translation is a slide. For a shape to return to its original position after a sequence of translations, the net displacement in each direction must be zero. In option A, 3 units up and 3 units down cancel each other out, but there is no information about horizontal displacement.

Step2: Analyze translation - reflection - translation

In option B, translating, reflecting, and then translating again does not guarantee a return to the original position as the reflection changes the orientation and the subsequent translation is not in a way to reverse all changes.

Step3: Analyze reflection - translation - reflection

In option C, reflecting over line \(q\), then translating, and reflecting over line \(p\) again. Reflections and translations do not necessarily cancel each other out in this combination to return the shape to its original position.

Step4: Analyze rotation - rotation

In option D, rotating \(120^{\circ}\) counter - clockwise and then \(240^{\circ}\) counter - clockwise around the same center \(C\). The sum of the rotation angles \(120^{\circ}+240^{\circ}=360^{\circ}\), which is a full - turn. A full - turn rotation around a center will return a shape to its original position.

Answer:

D