Question

the triangles are congruent by the sss congruence theorem.
which rigid transformation(s) can map △fgh onto △vwx?
options:
rotation, then dilation
rotation, then translation
reflection, then rotation
reflection, then translation

Explanation:

Step1: Recall Rigid Transformations

Rigid transformations (isometries) preserve shape and size, so dilation (which changes size) is not a rigid transformation. So "rotation, then dilation" is invalid.

Step2: Analyze Triangle Orientation

To map \( \triangle FGH \) onto \( \triangle VWX \), we can first reflect to align the orientation, then rotate to match the angle, or rotate then translate? Wait, rigid transformations include reflection, rotation, translation. Let's check the options:

• Option with dilation: invalid (dilation is not rigid).
• Now, between reflection/rotation and rotation/translation. The key is that after reflection (to flip orientation if needed) and then rotation, or rotation and translation. But since the triangles are congruent (SSS), rigid transformations apply. The correct rigid transformation sequence that works is "rotation, then translation" or "reflection, then rotation" or "reflection, then translation"? Wait, the option "rotation, then translation" is a valid rigid transformation sequence (both rotation and translation are rigid, preserve length and angle). Wait, let's re - check the options:

Wait the options are:

1. rotation, then dilation (invalid, dilation is not rigid)
2. reflection, then rotation (valid, both rigid)
3. rotation, then translation (valid, both rigid)

Wait, maybe I misread. Wait the problem says "which rigid transformation(s)". Wait, the triangles are congruent, so rigid transformations (which preserve congruence) are reflection, rotation, translation. Dilation is not rigid (changes size, so only for similar figures). So first, eliminate the option with dilation ("rotation, then dilation"). Now, between the other options. Let's look at the triangle positions. To map \( \triangle FGH \) to \( \triangle VWX \), we can rotate \( \triangle FGH \) to get the same angle orientation, then translate it to the position of \( \triangle VWX \). Or reflect then rotate. But among the given options, "rotation, then translation" is a valid rigid transformation sequence (since both rotation and translation are rigid, and they can map one congruent triangle to another). Wait, but let's check the options again. The option "rotation, then translation" is a valid rigid transformation (both preserve side lengths and angles). The other option with reflection and rotation is also valid, but maybe the intended answer is "rotation, then translation" or "reflection, then rotation"? Wait, no, let's think again.

Wait, the key is that dilation is not a rigid transformation, so we eliminate the option with dilation. Then, the remaining options: "reflection, then rotation" and "rotation, then translation" are both valid? Wait, no, maybe the diagram shows that after rotating \( \triangle FGH \) and then translating it, we can get to \( \triangle VWX \). So the correct option is "rotation, then translation" (since rotation and translation are rigid transformations, and dilation is not). Wait, but let's confirm:

Rigid transformations: reflection, rotation, translation. These preserve congruence. Dilation does not. So the option with dilation is out. Now, between the other two. Let's assume that the triangle can be rotated to align the sides and then translated to the correct position. So "rotation, then translation" is a valid rigid transformation sequence.

Answer:

The correct option is "rotation, then translation" (the option with rotation followed by translation, as dilation is not a rigid transformation and is excluded, and rotation then translation is a valid rigid transformation sequence to map \( \triangle FGH \) onto \( \triangle VWX \)).