Use the image to answer the question.

Question

Two acute scalene triangles of equal dimensions and the same orientation. Triangle upper K upper B upper P is in the upper left, and triangle upper A upper Q upper L is in the lower right.

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 B maps to point A
translation along the vector so that point upper B maps to point upper A

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

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 $PBK$ to triangle $LQA$, we need to analyze the relationship between the points in the two triangles:

1. Identify Corresponding Points:

   • It seems we want to match:
     • $P$ might correspond to $L$
     • $B$ might correspond to $Q$
     • $K$ might correspond to $A$

2. Evaluate Each Option:

• Translation along the vector so that point $B$ maps to point $A$: This option moves point $B$ directly to point $A$, but does not help to arrange the other vertices, $P$ and $K$.

• Translation along the vector so that point $P$ maps to point $A$: Moving $P$ to $A$ does not match the remaining vertices as $B$ and $Q$ would not align correctly.

• Rotation of 180° (counterclockwise) about point $B$: This option rotates $PBK$ around $B$. If $B$ functions as the axis of rotation, the other points do not align with $L$ and $A$.

• Translation along the vector so that point $B$ maps to point $Q$: This option translates $B$ directly to $Q$, which can then lead to verifying that $P$ will match with $L$ and $K$ with $A$. This translation aligns one pair of triangles properly.

Conclusion: The best option for mapping triangle $PBK$ to triangle $LQA$ is:

Translation along the vector so that point $B$ maps to point $Q$.