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 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 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° (counterclockwise) about point J
Answers
GPT-4o mini
To determine the correct rigid motion that would map triangle \( \triangle GHJ \) to triangle \( \triangle ZNP \), we need to analyze the options provided and the relationship between the vertices of the two triangles.
1. **Understanding the Rigid Motions**: Rigid motions include translations and rotations. A translation moves every point of a shape the same distance in a specified direction, while a rotation turns the shape around a fixed point (the center of rotation).
2. **Identifying Corresponding Points**: First, we need to establish how the points of triangle \( GHJ \) correspond to triangle \( ZNP \). Let's assume that:
- Point J corresponds to Point P,
- Point H corresponds to Point Z,
- Point G corresponds to Point N.
3. **Testing Rigid Motions**:
- **Translation along vector mapping J to P, then rotation -90° about J**: This would initially move J to P and then rotate the figure based on point J. This may not correctly map H to Z and G to N.
- **Translation along vector mapping G to N, then rotation -90° about G**: This would first move G to N and rotate the triangle about G. This could potentially work depending on the initial arrangement but is less likely since we need J to match with P.
- **Rotation -90° about J, then translation mapping H to Z**: This would rotate the triangle but might not initially align J and P correctly.
- **Translation mapping J to P, then rotation 90° (counterclockwise) about J**: This would map J to P and then rotate the rest of the triangle which could potentially align H to Z and G to N.
Considering these analyses, if we find that point J needs to align with point P, and the orientation is such that a translation followed by a 90-degree counterclockwise rotation about J could achieve this, then:
**The most likely correct option would be:**
**Translation along the vector, mapping point J to point P, then rotation 90° (counterclockwise) about point J.**
This preserves the orientation of the remaining points aligned accordingly to map \( \triangle GHJ \) to \( \triangle ZNP \).
1. **Understanding the Rigid Motions**: Rigid motions include translations and rotations. A translation moves every point of a shape the same distance in a specified direction, while a rotation turns the shape around a fixed point (the center of rotation).
2. **Identifying Corresponding Points**: First, we need to establish how the points of triangle \( GHJ \) correspond to triangle \( ZNP \). Let's assume that:
- Point J corresponds to Point P,
- Point H corresponds to Point Z,
- Point G corresponds to Point N.
3. **Testing Rigid Motions**:
- **Translation along vector mapping J to P, then rotation -90° about J**: This would initially move J to P and then rotate the figure based on point J. This may not correctly map H to Z and G to N.
- **Translation along vector mapping G to N, then rotation -90° about G**: This would first move G to N and rotate the triangle about G. This could potentially work depending on the initial arrangement but is less likely since we need J to match with P.
- **Rotation -90° about J, then translation mapping H to Z**: This would rotate the triangle but might not initially align J and P correctly.
- **Translation mapping J to P, then rotation 90° (counterclockwise) about J**: This would map J to P and then rotate the rest of the triangle which could potentially align H to Z and G to N.
Considering these analyses, if we find that point J needs to align with point P, and the orientation is such that a translation followed by a 90-degree counterclockwise rotation about J could achieve this, then:
**The most likely correct option would be:**
**Translation along the vector, mapping point J to point P, then rotation 90° (counterclockwise) about point J.**
This preserves the orientation of the remaining points aligned accordingly to map \( \triangle GHJ \) to \( \triangle ZNP \).