Question
Triangles H J K and L M N are congruent. Triangle H J K is rotated about point H to form triangle L N M. Triangle L M N is higher than triangle H J K.
How can a translation and a rotation be used to map ΔHJK to ΔLMN?
How can a translation and a rotation be used to map ΔHJK to ΔLMN?
Answers
GPT-4o mini
To map triangle \( \Delta HJK \) to triangle \( \Delta LMN \), you can follow these steps using a combination of a rotation and a translation:
1. **Identify the Rotation:**
Since it is given that triangle \( \Delta HJK \) is rotated about point \( H \) to form triangle \( \Delta LMN \), first determine the angle of rotation. The vertices of triangles \( HJK \) and \( LMN \) correspond as follows due to their congruence:
- \( H \) corresponds to \( L \)
- \( J \) corresponds to \( M \)
- \( K \) corresponds to \( N \)
You will need to rotate triangle \( HJK \) around point \( H \) to get point \( L \) at the correct position. Let's denote this rotation angle as \( \theta \).
2. **Perform the Rotation:**
Rotate triangle \( HJK \) by \( \theta \) degrees counterclockwise (or clockwise, depending on the situation) about point \( H \). This takes point \( J \) to point \( M \) and point \( K \) to point \( N \).
3. **Identify the Translation:**
After rotation, if the triangle \( \Delta HJK \) does not match completely with triangle \( \Delta LMN \) (i.e., if the position of points does not align perfectly), you will need to apply a translation.
Measure the distance and direction needed to translate the rotated triangle \( \Delta HJK \) so that \( L \) coincides with \( H \). This can involve moving the entire structure up or over while maintaining the angles.
4. **Perform the Translation:**
Translate the rotated triangle \( \Delta HJK \) by the result from your translation measurement to align it perfectly with triangle \( \Delta LMN \).
Combining these two steps—a rotation around point \( H \) followed by a translation—will map \( \Delta HJK \) onto \( \Delta LMN \).
In summary:
1. Rotate \( \Delta HJK \) around point \( H \) by the angle \( \theta \).
2. Translate the resulting triangle to ensure it coincides with \( \Delta LMN \).
This method guarantees that both triangles overlap perfectly, confirming their congruence.
1. **Identify the Rotation:**
Since it is given that triangle \( \Delta HJK \) is rotated about point \( H \) to form triangle \( \Delta LMN \), first determine the angle of rotation. The vertices of triangles \( HJK \) and \( LMN \) correspond as follows due to their congruence:
- \( H \) corresponds to \( L \)
- \( J \) corresponds to \( M \)
- \( K \) corresponds to \( N \)
You will need to rotate triangle \( HJK \) around point \( H \) to get point \( L \) at the correct position. Let's denote this rotation angle as \( \theta \).
2. **Perform the Rotation:**
Rotate triangle \( HJK \) by \( \theta \) degrees counterclockwise (or clockwise, depending on the situation) about point \( H \). This takes point \( J \) to point \( M \) and point \( K \) to point \( N \).
3. **Identify the Translation:**
After rotation, if the triangle \( \Delta HJK \) does not match completely with triangle \( \Delta LMN \) (i.e., if the position of points does not align perfectly), you will need to apply a translation.
Measure the distance and direction needed to translate the rotated triangle \( \Delta HJK \) so that \( L \) coincides with \( H \). This can involve moving the entire structure up or over while maintaining the angles.
4. **Perform the Translation:**
Translate the rotated triangle \( \Delta HJK \) by the result from your translation measurement to align it perfectly with triangle \( \Delta LMN \).
Combining these two steps—a rotation around point \( H \) followed by a translation—will map \( \Delta HJK \) onto \( \Delta LMN \).
In summary:
1. Rotate \( \Delta HJK \) around point \( H \) by the angle \( \theta \).
2. Translate the resulting triangle to ensure it coincides with \( \Delta LMN \).
This method guarantees that both triangles overlap perfectly, confirming their congruence.