Question
To prove the triangles are congruent, which of the following rigid motions would map △GHJ to △ZNP ? (1 point) Responses 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 translation along the vector, mapping point upper J to point upper P , then rotation negative 90 degrees (clockwise) about point upper J 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 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
Answers
GPT-4o mini
To determine which rigid motion maps triangle GHJ to triangle ZNP, analyze the relationships between the triangles' points.
1. **Identify Key Points:**
- For triangle GHJ: Points are G, H, J.
- For triangle ZNP: Points are Z, N, P.
2. **Establish Rigid Motion Steps:**
- Remember that rigid motions include translations, rotations, and reflections. We need to check potential movements that align corresponding points of the triangles.
3. **Evaluate Each Option:**
- **Option 1**: Rotate -90° clockwise about point J, then translate mapping H to Z.
- **Option 2**: Rotate -90° clockwise about point J, then translate mapping H to Z. *Identical to Option 1, but noted differently.*
- **Option 3**: Translate J to P, then rotate -90° clockwise about J.
- **Option 4**: Translate J to P, then rotate -90° clockwise about J. *Identical to Option 3, but noted differently.*
- **Option 5**: Translate J to P, then rotate 90° counterclockwise about J.
- **Option 6**: Translate J to P, then rotate 90° counterclockwise about J. *Identical to Option 5, but noted differently.*
- **Option 7**: Translate G to N, then rotate -90° clockwise about G.
- **Option 8**: Translate G to N, then rotate -90° clockwise about G. *Identical to Option 7, but noted differently.*
4. **Choosing the Right Option:**
- If we consider the necessary rotations and the points involved, options utilizing translation and rotation from J to P will likely yield the necessary alignment for congruence with ZNP.
Based on the analysis, the suitable choice would likely involve either directly translating J to P and applying a clockwise rotation around J or G.
**Best Options:**
- **Translate along the vector, mapping point J to point P, then rotate -90° (clockwise) about point J.** This option would align the triangles correctly through the specified rigid motions.
Thus, the option is:
**Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J.**
1. **Identify Key Points:**
- For triangle GHJ: Points are G, H, J.
- For triangle ZNP: Points are Z, N, P.
2. **Establish Rigid Motion Steps:**
- Remember that rigid motions include translations, rotations, and reflections. We need to check potential movements that align corresponding points of the triangles.
3. **Evaluate Each Option:**
- **Option 1**: Rotate -90° clockwise about point J, then translate mapping H to Z.
- **Option 2**: Rotate -90° clockwise about point J, then translate mapping H to Z. *Identical to Option 1, but noted differently.*
- **Option 3**: Translate J to P, then rotate -90° clockwise about J.
- **Option 4**: Translate J to P, then rotate -90° clockwise about J. *Identical to Option 3, but noted differently.*
- **Option 5**: Translate J to P, then rotate 90° counterclockwise about J.
- **Option 6**: Translate J to P, then rotate 90° counterclockwise about J. *Identical to Option 5, but noted differently.*
- **Option 7**: Translate G to N, then rotate -90° clockwise about G.
- **Option 8**: Translate G to N, then rotate -90° clockwise about G. *Identical to Option 7, but noted differently.*
4. **Choosing the Right Option:**
- If we consider the necessary rotations and the points involved, options utilizing translation and rotation from J to P will likely yield the necessary alignment for congruence with ZNP.
Based on the analysis, the suitable choice would likely involve either directly translating J to P and applying a clockwise rotation around J or G.
**Best Options:**
- **Translate along the vector, mapping point J to point P, then rotate -90° (clockwise) about point J.** This option would align the triangles correctly through the specified rigid motions.
Thus, the option is:
**Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J.**
sora
abcd format
GPT-4o mini
Sure! Based on the analysis of the motions needed to map triangle GHJ to triangle ZNP, here is the response in the requested format (abcd):
**a.** Rotation -90° clockwise about point J, then translation along the vector, mapping point H to point Z.
**b.** Rotation -90° clockwise about point J, then translation along the vector, mapping point H to point Z.
**c.** Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J.
**d.** Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J.
**e.** Translation along the vector, mapping point J to point P, then rotation 90° (counterclockwise) about point J.
**f.** Translation along the vector, mapping point J to point P, then rotation 90° (counterclockwise) about point J.
**g.** Translation along the vector, mapping point G to point N, then rotation -90° (clockwise) about point G.
**h.** Translation along the vector, mapping point G to point N, then rotation -90° (clockwise) about point G.
### Correct Option:
**Option c** (or d) would be the right choice:
**c.** Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J.
**a.** Rotation -90° clockwise about point J, then translation along the vector, mapping point H to point Z.
**b.** Rotation -90° clockwise about point J, then translation along the vector, mapping point H to point Z.
**c.** Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J.
**d.** Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J.
**e.** Translation along the vector, mapping point J to point P, then rotation 90° (counterclockwise) about point J.
**f.** Translation along the vector, mapping point J to point P, then rotation 90° (counterclockwise) about point J.
**g.** Translation along the vector, mapping point G to point N, then rotation -90° (clockwise) about point G.
**h.** Translation along the vector, mapping point G to point N, then rotation -90° (clockwise) about point G.
### Correct Option:
**Option c** (or d) would be the right choice:
**c.** Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J.