Question

To prove the triangles are congruent, which of the following rigid motions would map △PBK to △LQA ? (1 point) Responses translation along the vector so that point P maps to point A translation along the vector so that point upper P maps to point upper A translation along the vector so that point B maps to point A translation along the vector so that point upper B maps to point upper A rotation of 180° (counterclockwise) about point B rotation of 180 degrees (counterclockwise) about point upper B translation along the vector so that point B maps to point Q translation along the vector so that point upper B maps to point upper Q

Answer 1

To determine which rigid motion would map triangle \( \triangle PBK \) to triangle \( \triangle LQA \), we need to analyze the correspondence between the vertices of the two triangles and the types of rigid motions available.

Assuming the points map as follows:
- \( P \) to \( L \)
- \( B \) to \( Q \)
- \( K \) to \( A \)

Now, let's analyze the options provided:

1. **Translation along the vector so that point P maps to point A** - This does not match the mapping of \( P \) to \( L \).

2. **Translation along the vector so that point P maps to point A** - Same reason as above.

3. **Translation along the vector so that point B maps to point A** - This does not match as we expect \( B \) to map to \( Q \).

4. **Translation along the vector so that point B maps to point A** - Same reason as above.

5. **Rotation of 180° (counterclockwise) about point B** - This could change the positions of points but does not fit our necessary mapping.

6. **Rotation of 180 degrees (counterclockwise) about point B** - Similar reasoning as above.

7. **Translation along the vector so that point B maps to point Q** - This is a likely candidate since it directly states that \( B \) would go to \( Q \).

8. **Translation along the vector so that point B maps to point Q** - Same as above.

From the analysis, the option **translation along the vector so that point B maps to point Q** is the best candidate since we can then find if subsequent translations or rotations can adjust the positions of \( P \) to \( L \) and \( K \) to \( A \).

Thus, the correct rigid motion to prove the triangles are congruent is **translation along the vector so that point B maps to point Q**.