Question

To prove the triangles are congruent, which of the following rigid motions would map △GHJ to △ZNP ?

(1 point)
Responses

translation along the vector, mapping point G to point N, then rotation −90° (clockwise) about point G
translation along the vector, mapping point upper G to point upper N , then rotation negative 90 degrees (clockwise) about point upper G

translation along the vector, mapping point J to point P, then rotation 90° (counterclockwise) about point J
translation along the vector, mapping point upper J to point upper P , then rotation 90 degrees (counterclockwise) about point upper J

rotation −90° (clockwise) about point J, then translation along the vector, mapping point H to point Z
rotation negative 90 degrees (clockwise) about point upper J , then translation along the vector, mapping point upper H to point upper Z

translation along the vector, mapping point J to point P, then rotation −90° (clockwise) about point J

Answer 1

To determine which rigid motion maps triangle \( \triangle GHJ \) to triangle \( \triangle ZNP \), we need to analyze the sequences of transformations provided in the options.

1. **Going through the options:**
- **Option 1:** Translation mapping \( G \) to \( N \) followed by a rotation \( -90^\circ \) (clockwise) about \( G \).
- **Option 2:** Same as Option 1 but written differently.
- **Option 3:** Translation mapping \( J \) to \( P \) followed by a rotation \( 90^\circ \) (counterclockwise) about \( J \).
- **Option 4:** Same as Option 3 but written differently.
- **Option 5:** Rotation \( -90^\circ \) (clockwise) about \( J \) then translation mapping \( H \) to \( Z \).
- **Option 6:** Rotation \( -90^\circ \) (clockwise) about \( J \) then translation mapping \( H \) to \( Z \).
- **Option 7:** Translation mapping \( J \) to \( P \) then a rotation \( -90^\circ \) (clockwise) about \( J \).

2. **Understanding rigid motions:**
- Rigid motions include translations and rotations that preserve the shape and size of triangles.
- When mapping one triangle to another, it is essential to match corresponding points correctly (i.e., \( G \) to \( Z \), \( H \) to \( N \), and \( J \) to \( P \)).

3. **Analyzing from a geometric perspective:**
- If translating point \( J \) to point \( P \) maintains the integrity of corresponding sides and angles with respect to rotation, this is likely a viable option.
- The necessary rotation afterward can help fit the angles of triangle \( GHJ \) to triangle \( ZNP \) due to their respective orientations.

4. **Final match:**
- Since it makes sense for the transformation to start with a translation of point \( J \) to point \( P \), any options that follow this action would be considered, focusing on how this could lead to proper orientation via rotation.
- Among the options, **Option 7** fits this requirement since it begins with translating \( J \) to \( P \), on top of allowing the fitting rotation of \( -90^\circ \) (clockwise) around \( J \), aligning effectively to correspond the triangles' configurations.

Thus, the correct answer is the option:

**Translation along the vector, mapping point \( J \) to point \( P \), then rotation \( -90^\circ \) (clockwise) about point \( J \)**.