Which transformations or sequence of transformations prove the objects are congruent? Select THREE that are correct.
Translate Object 1 by –4 units along the x-axis, and then translate it by –12 along the y-axis.
Translate Object 1 by –12 units along the x-axis, and then translate it by –4 along the y-axis.
Reflect Object 1 over the x-axis, and then translate it by –13 along the x-axis.
Rotate Object 1 by 180° clockwise about the origin, and then translate it by −5 along the x-axis.
Rotate Object 1 by 180° counterclockwise about the origin, and then translate it by −5 along the x-axis.
Reflect Object 1 over the y-axis, and then translate it by −13 along the x-axis.
Translate Object 1 by –4 units along the x-axis, and then translate it by –12 along the y-axis.
Reflect Object 1 over the x-axis, and then translate it by –13 along the x-axis.
Rotate Object 1 by 180° counterclockwise about the origin, and then translate it by −5 along the x-axis.
To determine which transformations or sequence of transformations prove the objects are congruent, we need to understand the concept of congruence. Two objects are congruent if they have the same size and shape.
Let's analyze each option:
1. Translate Object 1 by –4 units along the x-axis, and then translate it by –12 along the y-axis.
This transformation involves two translations, first along the x-axis and then along the y-axis. Since translations preserve both size and shape, this transformation alone proves the objects are congruent.
2. Translate Object 1 by –12 units along the x-axis, and then translate it by –4 along the y-axis.
Similar to the first option, this transformation also involves two translations. However, the order of the translations is different. Nevertheless, translations preserve congruence, so this transformation also proves the objects are congruent.
3. Reflect Object 1 over the x-axis, and then translate it by –13 along the x-axis.
A reflection over the x-axis changes the orientation of an object, but it preserves its size and shape. However, combining it with a subsequent translation along the x-axis can change the position of the object. Therefore, this transformation alone does not prove congruence.
4. Rotate Object 1 by 180° clockwise about the origin, and then translate it by −5 along the x-axis.
Rotation by 180° about the origin preserves size and shape, as long as it is 180° clockwise or counterclockwise. The subsequent translation only changes the position, so this transformation alone does not prove congruence.
5. Rotate Object 1 by 180° counterclockwise about the origin, and then translate it by −5 along the x-axis.
Similar to the previous option, rotation by 180° counterclockwise about the origin preserves size and shape. The subsequent translation only changes the position, so this transformation alone does not prove congruence.
6. Reflect Object 1 over the y-axis, and then translate it by −13 along the x-axis.
A reflection over the y-axis, like a reflection over the x-axis, changes the orientation of an object but preserves its size and shape. Combining it with a subsequent translation along the x-axis can change the position of the object. Therefore, this transformation alone does not prove congruence.
Based on the analysis, the three correct options that prove the objects are congruent are:
1. Translate Object 1 by –4 units along the x-axis, and then translate it by –12 along the y-axis.
2. Translate Object 1 by –12 units along the x-axis, and then translate it by –4 along the y-axis.
3. Reflect Object 1 over the x-axis, and then translate it by –13 along the x-axis.
The correct transformations that prove the objects are congruent are:
1. Translate Object 1 by –4 units along the x-axis, and then translate it by –12 along the y-axis.
2. Translate Object 1 by –12 units along the x-axis, and then translate it by –4 along the y-axis.
3. Reflect Object 1 over the y-axis, and then translate it by −13 along the x-axis.