Question

congruency of triangles quick check
use the image to answer the question
to prove the triangles are congruent, which of the following rigid motions would map △ghj to △znp?
(1 point)
○ translation along the vector, mapping point g to point n, then rotation - 90° (clockwise) about point g
○ translation along the vector, mapping point j to point p, then rotation 90° (counterclockwise) about point j
○ translation along the vector, mapping point j to point p, then rotation - 90° (clockwise) about point j
○ rotation - 90° (clockwise) about point j, then translation along the vector, mapping point h to point z

Explanation:

Step1: Analyze the order of transformation

Rigid motions include translation and rotation. Translation moves the figure without rotation, and rotation turns the figure around a point. For congruent triangles, the correct sequence of transformation should map corresponding vertices.

Step2: Check each option

• Option 1: Translation mapping \(G\) to \(N\) then rotation about \(G\). But after translation, rotation about \(G\) (which is now at \(N\)) won't align the triangles properly as the orientation and position - to - vertex correspondence is wrong.
• Option 2: Translation mapping \(J\) to \(P\) then rotation \(90^{\circ}\) counter - clockwise about \(J\) (now at \(P\)). This will not align the triangles as the rotation direction and the subsequent orientation won't match the target triangle \(\triangle ZNP\).
• Option 3: Translation mapping \(J\) to \(P\) (moves the triangle so that one vertex is in place). Then rotation \(- 90^{\circ}\) (clockwise) about \(J\) (now at \(P\)). This rotation will align the other vertices (\(H\) and \(G\) of \(\triangle GHJ\)) with \(Z\) and \(N\) of \(\triangle ZNP\) respectively.
• Option 4: Rotation about \(J\) first then translation. The order is wrong because if we rotate first, the position of the triangle before translation is not in a state that can be simply translated to match \(\triangle ZNP\) as the vertex - to - vertex correspondence (due to the initial rotation) is off.

Answer:

translation along the vector, mapping point \(J\) to point \(P\), then rotation \(-90^{\circ}\) (clockwise) about point \(J\)