Question
To prove the triangles are congruent, which of the following rigid motions would map △GHJ to △ZNP ?
Possible Answers:
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° (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 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° (counterclockwise) about point J
Possible Answers:
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° (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 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° (counterclockwise) about point J
Answers
GPT-4o mini
To determine which rigid motions would map triangle ∆GHJ to triangle ∆ZNP, we need to analyze the correspondence of the vertices and the potential transformations.
1. **Identify the Correspondence of Vertices**:
- G → N
- H → Z
- J → P
2. **Check for Rigid Motions**:
- **Translations**: We can translate points in the plane, but we must also ensure that the orientation and distance are preserved afterward.
- **Rotations**: These involve spinning the triangle around a point, which can change the orientation.
3. **Analyze Options**:
- A **translation along the vector** mapping G to N and then a **rotation -90° (clockwise)** about G. This would not correctly rotate H and J to their corresponding points.
- A **translation along the vector** mapping J to P and then a **rotation -90° (clockwise)** about J does not help in addressing the mapping of point G to point N.
- A **rotation -90° (clockwise)** about J followed by a **translation mapping H to Z** could be worth checking.
- A **translation mapping J to P** followed by a **rotation 90° (counterclockwise)** about J is another candidate.
4. **Evaluation**:
- For a correct mapping:
- Map G to N (translation)
- Map H to Z (rotation and reference point consideration)
- Map J to P (translation or rotation relation to H)
Among the listed options, based on logical deductions and evaluations:
- The option "rotation -90° (clockwise) about point J, then translation along the vector, mapping point H to point Z" can provide a likely transformation sequence. This allows you to first position J correctly and rotate the triangle around point J, then align the point H to the right position afterward.
Thus, this reasoning points toward:
**Option: rotation -90° (clockwise) about point J, then translation along the vector, mapping point H to Z.**
This option effectively corresponds to the necessary transformations to map triangle ∆GHJ onto triangle ∆ZNP.
1. **Identify the Correspondence of Vertices**:
- G → N
- H → Z
- J → P
2. **Check for Rigid Motions**:
- **Translations**: We can translate points in the plane, but we must also ensure that the orientation and distance are preserved afterward.
- **Rotations**: These involve spinning the triangle around a point, which can change the orientation.
3. **Analyze Options**:
- A **translation along the vector** mapping G to N and then a **rotation -90° (clockwise)** about G. This would not correctly rotate H and J to their corresponding points.
- A **translation along the vector** mapping J to P and then a **rotation -90° (clockwise)** about J does not help in addressing the mapping of point G to point N.
- A **rotation -90° (clockwise)** about J followed by a **translation mapping H to Z** could be worth checking.
- A **translation mapping J to P** followed by a **rotation 90° (counterclockwise)** about J is another candidate.
4. **Evaluation**:
- For a correct mapping:
- Map G to N (translation)
- Map H to Z (rotation and reference point consideration)
- Map J to P (translation or rotation relation to H)
Among the listed options, based on logical deductions and evaluations:
- The option "rotation -90° (clockwise) about point J, then translation along the vector, mapping point H to point Z" can provide a likely transformation sequence. This allows you to first position J correctly and rotate the triangle around point J, then align the point H to the right position afterward.
Thus, this reasoning points toward:
**Option: rotation -90° (clockwise) about point J, then translation along the vector, mapping point H to Z.**
This option effectively corresponds to the necessary transformations to map triangle ∆GHJ onto triangle ∆ZNP.