Question

1. triangle abc is congruent to triangle abc.

describe a sequence of rigid motions that takes a to a, b to b, and c to c.

Explanation:

Step1: Translate triangle ABC

First, translate triangle \(ABC\) so that point \(A\) coincides with point \(A'\). Let the translation vector be \(\overrightarrow{AA'}\). After translation, the new position of triangle \(ABC\) (let's call it \(A''B''C''\)) has \(A'' = A'\).

Step2: Rotate the translated triangle

Then, rotate the translated triangle \(A''B''C''\) around point \(A'\) (since \(A''=A'\)) until \(B''\) coincides with \(B'\). The rotation angle is the angle between \(\overrightarrow{A''B''}\) (now \(\overrightarrow{A'B''}\)) and \(\overrightarrow{A'B'}\). After this rotation, since the triangles are congruent (\(AB = A'B'\), \(BC=B'C'\), \(AC = A'C'\)), point \(C''\) will also coincide with \(C'\).

Answer:

First, translate triangle \(ABC\) by the vector \(\overrightarrow{AA'}\). Then, rotate the translated triangle around point \(A'\) by the angle between \(\overrightarrow{A'B''}\) (where \(B''\) is the image of \(B\) after translation) and \(\overrightarrow{A'B'}\) so that \(B\) maps to \(B'\) and \(C\) maps to \(C'\)