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 p
(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 p
Answers
GPT-4o mini
To determine the correct rigid motions that can map triangle \( \triangle GHJ \) to triangle \( \triangle ZNP \), we need to consider the relationships between the corresponding vertices of both triangles and the sequence of rigid transformations.
When working with triangles and rigid motions (translations, rotations), we look for a combination that will preserve distance and angles, thus ensuring that triangles remain congruent.
1. **Identify corresponding vertices**: Start by mapping vertices of triangle \( \triangle GHJ \) to their corresponding points in \( \triangle ZNP \). From what's given:
- \(G\) should map to \(Z\),
- \(H\) should map to \(N\),
- \(J\) should map to \(P\).
2. **Choose a rigid motion**: We may choose to either perform a translation or a rotation first. The order can change based on how the points correlate.
3. **Assess possible motions**:
- **Translation along the vector, mapping \(G\) to \(N\)**: This suggests moving point \(G\) to point \(N\) first. However, since we established that \(G\) should eventually correspond to \(Z\), this might not be a valid starting point.
- **Rotation -90° about a point**: This would only be applicable if the angle positioning matches with the corresponding angles in triangle \(ZNP\).
4. **After analyzing**: Among the options given in your inquiry, a typical sequence following rigid transformations typically would look like:
- **Translation of \(J\) to \(P\)** — This would correctly place vertex \(J\) according to our requirement.
- **Then a rotation clockwise (or counterclockwise) as necessary** should keep \(GHJ\) congruent to \(ZNP\).
5. **Final determination**: Given that one of the options states, “translation along the vector, mapping point J to point P, then rotation −90° (clockwise) about point J”, this choice would effectively maintain the vertices and congruence, mapping triangle \(GHJ \) correctly onto triangle \(ZNP\).
So the correct response is:
**translation along the vector, mapping point J to point P, then rotation −90° (clockwise) about point J.**
When working with triangles and rigid motions (translations, rotations), we look for a combination that will preserve distance and angles, thus ensuring that triangles remain congruent.
1. **Identify corresponding vertices**: Start by mapping vertices of triangle \( \triangle GHJ \) to their corresponding points in \( \triangle ZNP \). From what's given:
- \(G\) should map to \(Z\),
- \(H\) should map to \(N\),
- \(J\) should map to \(P\).
2. **Choose a rigid motion**: We may choose to either perform a translation or a rotation first. The order can change based on how the points correlate.
3. **Assess possible motions**:
- **Translation along the vector, mapping \(G\) to \(N\)**: This suggests moving point \(G\) to point \(N\) first. However, since we established that \(G\) should eventually correspond to \(Z\), this might not be a valid starting point.
- **Rotation -90° about a point**: This would only be applicable if the angle positioning matches with the corresponding angles in triangle \(ZNP\).
4. **After analyzing**: Among the options given in your inquiry, a typical sequence following rigid transformations typically would look like:
- **Translation of \(J\) to \(P\)** — This would correctly place vertex \(J\) according to our requirement.
- **Then a rotation clockwise (or counterclockwise) as necessary** should keep \(GHJ\) congruent to \(ZNP\).
5. **Final determination**: Given that one of the options states, “translation along the vector, mapping point J to point P, then rotation −90° (clockwise) about point J”, this choice would effectively maintain the vertices and congruence, mapping triangle \(GHJ \) correctly onto triangle \(ZNP\).
So the correct response is:
**translation along the vector, mapping point J to point P, then rotation −90° (clockwise) about point J.**