Question

6 from unit 1, lesson 17
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 \(ABC\)

Translate triangle \(ABC\) so that point \(A\) coincides with point \(A'\). Let the translation vector be \(\overrightarrow{AA'}\). After translation, the image of \(A\) is \(A'\), the image of \(B\) is \(B_1\) and the image of \(C\) is \(C_1\).

Step2: Rotate the translated triangle

Rotate the translated triangle (with \(A = A'\)) around point \(A'\) until \(B_1\) coincides with \(B'\). Let the angle of rotation be \(\theta=\angle B_1A'B'\). After rotation, since the triangles are congruent (\(AB = A'B'\), \(AC=A'C'\), \(BC = B'C'\)), the image of \(C_1\) will coincide with \(C'\)

Answer:

First, translate triangle \(ABC\) by the vector \(\overrightarrow{AA'}\). Then, rotate the translated triangle around point \(A'\) by the angle that maps the translated - image of \(B\) to \(B'\) (this rotation will also map the translated - image of \(C\) to \(C'\))