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 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
Answers
GPT-4o mini
To determine which of the given rigid motions will map triangle \( GHJ \) onto triangle \( ZNP \), we need to analyze the options provided. We'll look for sequences that involve translations and rotations that maintain the congruence and orientation needed to match the vertices of \( GHJ \) to \( ZNP \).
1. **Translation along the vector, mapping point G to point N, then rotation -90° (clockwise) about point G**: This would move point G to point N, and then rotate the triangle. This may not necessarily map the other points correctly.
2. **Translation along the vector, mapping point G to point N , then rotation -90° (clockwise) about point G**: This is identical to the first option, and has the same potential issues.
3. **Rotation -90° (clockwise) about point J, then translation along the vector, mapping point H to point Z**: This option first rotates the triangle, which could reposition the vertices. If point H ends up in the right position relative to Z, this might work.
4. **Rotation -90° (clockwise) about point J, then translation along the vector, mapping point H to point Z**: This is the same as the previous option, assuming it’s supposed to be a repeat.
5. **Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J**: This option translates J to P and then rotates the triangle. We would need to analyze if this could correctly position the remaining points.
6. **Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J**: This is a repeat of the 5th option.
7. **Translation along the vector, mapping J to P, then rotation 90° (counterclockwise) about point J**: This option involves a counterclockwise rotation after translating J to P.
Given these options, we will choose the second option, which seems to allow for a rotation that can align points after translating. Depending on the precise geometry of the triangles involved, the best choice for mapping must be tested.
However, to select a definitive option, more information about the positioning or coordinates of the points would be helpful.
Therefore, **the most effective mapping would be**:
- **Translation along the vector mapping point J to point P, then rotation -90° (clockwise) about point J** (the fifth/ sixth option).
This sequence of transformations appears likely to correctly map the original triangle to the target triangle.
1. **Translation along the vector, mapping point G to point N, then rotation -90° (clockwise) about point G**: This would move point G to point N, and then rotate the triangle. This may not necessarily map the other points correctly.
2. **Translation along the vector, mapping point G to point N , then rotation -90° (clockwise) about point G**: This is identical to the first option, and has the same potential issues.
3. **Rotation -90° (clockwise) about point J, then translation along the vector, mapping point H to point Z**: This option first rotates the triangle, which could reposition the vertices. If point H ends up in the right position relative to Z, this might work.
4. **Rotation -90° (clockwise) about point J, then translation along the vector, mapping point H to point Z**: This is the same as the previous option, assuming it’s supposed to be a repeat.
5. **Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J**: This option translates J to P and then rotates the triangle. We would need to analyze if this could correctly position the remaining points.
6. **Translation along the vector, mapping point J to point P, then rotation -90° (clockwise) about point J**: This is a repeat of the 5th option.
7. **Translation along the vector, mapping J to P, then rotation 90° (counterclockwise) about point J**: This option involves a counterclockwise rotation after translating J to P.
Given these options, we will choose the second option, which seems to allow for a rotation that can align points after translating. Depending on the precise geometry of the triangles involved, the best choice for mapping must be tested.
However, to select a definitive option, more information about the positioning or coordinates of the points would be helpful.
Therefore, **the most effective mapping would be**:
- **Translation along the vector mapping point J to point P, then rotation -90° (clockwise) about point J** (the fifth/ sixth option).
This sequence of transformations appears likely to correctly map the original triangle to the target triangle.