Question

to prove the triangles are congruent, which of the following rigid motions would map \\( \triangle pbk \\) to \\( \triangle lqa \\)? (1 point) translation along the vector so that point \\( p \\) maps to point \\( a \\) rotation of \\( 180^{circ} \\) (counterclockwise) about point \\( b \\) translation along the vector so that point \\( b \\) maps to point \\( q \\) translation along the vector so that point \\( b \\) maps to point \\( a \\)

Explanation:

Step1: Understand translation

Translation moves every point of a figure or space by the same distance in a given direction.

Step2: Analyze each option

• Option 1: If we translate along the vector so that point \(P\) maps to point \(A\), the other points \(B\) and \(K\) will not map to \(Q\) and \(L\) respectively.
• Option 2: Rotation of \(180^{\circ}\) about point \(B\) will not map \(\triangle PBK\) to \(\triangle LQA\) as the orientation and position of non - \(B\) related points will not match.
• Option 3: If we translate along the vector so that point \(B\) maps to point \(Q\), then \(P\) will map to \(L\) and \(K\) will map to \(A\) (because in a translation, the entire figure moves with the same vector).
• Option 4: If we translate along the vector so that point \(B\) maps to point \(A\), the other points \(P\) and \(K\) will not map to \(L\) and \(Q\) respectively.

Answer:

translation along the vector so that point \(B\) maps to point \(Q\)