Question

1. math talk: transformations

each pair of shapes is congruent. identify a transformation or sequence of transformations that could take one shape to the other.

Explanation:

Step1: Analyze circles

For the circles with centers \(P\) and \(Q\), a translation can move the circle centered at \(P\) to the circle centered at \(Q\). We can find the vector from the center of the first - circle to the center of the second - circle and translate the first circle along that vector.

Step2: Analyze line - segments

For line - segments \(\overline{AB}\) and \(\overline{CD}\), a translation can be used. First, find the vector from point \(A\) to point \(C\) (or from point \(B\) to point \(D\)). Translating \(\overline{AB}\) along this vector will map it onto \(\overline{CD}\).

Step3: Analyze triangles

For \(\triangle DEF\) and \(\triangle KLJ\), a rotation followed by a translation may be needed. First, find the center of rotation (it could be a point such that when \(\triangle DEF\) is rotated around it, its orientation matches that of \(\triangle KLJ\)). Then, a translation is applied to move the rotated \(\triangle DEF\) onto \(\triangle KLJ\).

Step4: Analyze line - segments (second pair)

For line - segments \(\overline{ST}\) and \(\overline{MN}\), a rotation and a translation are required. First, rotate \(\overline{ST}\) around a point to make its orientation the same as \(\overline{MN}\). Then, translate the rotated \(\overline{ST}\) to coincide with \(\overline{MN}\).

Answer:

For the circles: Translation. For the first pair of line - segments: Translation. For the triangles: Rotation followed by translation. For the second pair of line - segments: Rotation followed by translation.