Triangle $ABC$ has vertices $A$, $B$, and $C$. Under which transformation(s) will the length $AB$ remain equal to $A\text{'}B\text{'}$ after the transformation?

• A. Dilations (scalings) with a scale factor not equal to $1$
• B. Rigid motions such as translations, rotations, and reflections
• C. Only translations
• D. Shears

Answer: (B)

Verified step by step guidance

1. Understand that the length of a segment between two points is preserved only under transformations that do not change distances, called rigid motions.
2. Recall that rigid motions include translations (sliding the figure), rotations (turning the figure), and reflections (flipping the figure), all of which preserve lengths.
3. Recognize that dilations (scalings) with a scale factor not equal to 1 change the size of the figure, thus changing the length of segment AB.
4. Note that shears distort the shape by slanting it, which alters lengths and angles, so segment AB's length will not remain the same.
5. Conclude that only rigid motions—translations, rotations, and reflections—will keep the length AB equal to A'B' after the transformation.