Question

given △qrs ≅ △qrs, describe a pair of rigid motions that maps △qrs to △qrs:

Explanation:

Step1: Observe the orientation change

The triangle needs to be rotated 180° about the origin to change its orientation.

Step2: Observe the position change

After rotation, it needs to be translated. The translation vector can be found by comparing the position of a corresponding vertex before and after the transformation. For example, if we consider vertex Q, after rotating 180° about the origin, we then translate the triangle so that the rotated - Q coincides with Q'. A possible pair of rigid motions is a 180° rotation about the origin followed by a translation. Let's assume the coordinates of the vertices of \(\triangle QRS\) are \(Q(x_1,y_1)\), \(R(x_2,y_2)\), \(S(x_3,y_3)\) and of \(\triangle Q'R'S'\) are \(Q'(x_4,y_4)\), \(R'(x_5,y_5)\), \(S'(x_6,y_6)\). After rotation of point \((x,y)\) 180° about the origin, the new point is \((-x,-y)\). Then we find the translation vector \((a,b)\) such that \((-x_1 + a,y_1 + b)=x_4,y_4\).

Answer:

A 180° rotation about the origin followed by a translation.