Question

triangle opq and triangle jkl are drawn below. if \\(\overline{op} \cong \overline{jk}\\), \\(\overline{pq} \cong \overline{kl}\\), and \\(\overline{oq} \cong \overline{jl}\\), use the dropdown boxes below to determine a sequence of two transformations that maps triangle opq onto triangle jkl. then use a congruence statement to explain why this is possible. rotate triangle opq counterclockwise about point o until op has the same slope as jk. then translate triangle opq until point aligns with point.

Explanation:

Step1: Rotate the triangle

Rotate triangle \(OPQ\) counter - clockwise about point \(O\) until \(\overline{OP}\) has the same slope as \(\overline{JK}\). This rotation will adjust the orientation of the triangle \(OPQ\) so that one of its sides (\(\overline{OP}\)) is in a similar orientation as a corresponding side (\(\overline{JK}\)) of triangle \(JKL\).

Step2: Translate the triangle

Then translate triangle \(O'P'Q'\) until point \(O'\) aligns with point \(J\). Since \(\overline{OP}\cong\overline{JK}\), \(\overline{PQ}\cong\overline{KL}\), and \(\overline{OQ}\cong\overline{JL}\), by the Side - Side - Side (SSS) congruence criterion \(\triangle OPQ\cong\triangle JKL\). When we perform the rotation and translation (which are rigid motions), the two congruent triangles will map onto each other. Rigid motions (rotations and translations) preserve the shape and size of a figure.

Answer:

First, rotate triangle \(OPQ\) counter - clockwise about point \(O\) until \(\overline{OP}\) has the same slope as \(\overline{JK}\). Then translate triangle \(O'P'Q'\) until point \(O'\) aligns with point \(J\). The congruence statement is \(\triangle OPQ\cong\triangle JKL\) (by SSS), which means that a sequence of rigid motions (rotation and translation) can map \(\triangle OPQ\) onto \(\triangle JKL\) because rigid motions preserve congruence.