Question

the segment qp can be mapped to segment <br>by any composition of rigid motions. complete the statement below to prove that △tsu is congruent to △qpr.

Explanation:

Step1: Recall rigid - motion properties

Rigid motions (translations, rotations, reflections) preserve distance. For two segments to be mappable to each other by rigid - motions, they must have the same length.

Step2: Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Let's assume the coordinates of the endpoints of segment \(QP\) are \((x_{Q},y_{Q})\) and \((x_{P},y_{P})\), and for segment \(TS\) are \((x_{T},y_{T})\) and \((x_{S},y_{S})\). Calculate the lengths of the segments. If the lengths are equal, and we can find a combination of translations, rotations, and reflections to map one segment onto the other.

Step3: Analyze congruent triangles

Since we want to prove \(\triangle TSU\cong\triangle QPR\), corresponding sides must be congruent. Segment \(QP\) corresponds to segment \(TS\) in the congruence of the triangles. If \(\triangle TSU\cong\triangle QPR\), then segment \(QP\) can be mapped to segment \(TS\) by a composition of rigid motions.

Answer:

TS