Question

determining if figures are congruent
are rectangles abcd and wxyz congruent? why or why not?
yes, both rectangles have an area of 24 square
units.
yes, a 180° rotation about the origin will map
abcd to wxyz.
yes, all right angles were preserved.
no, there is not a rigid transformation or a
combination of rigid transformations that will map

Explanation:

Step1: Recall the definition of congruent figures

Congruent figures have the same shape and size. Rigid transformations (translations, rotations, reflections) preserve the shape and size of a figure.

Step2: Analyze the side - lengths of the rectangles

Let's assume the side - lengths of rectangle \(ABCD\). If we count the units on the coordinate grid, for rectangle \(ABCD\), let the length \(AB\) and \(CD\) be \(l_1\) and the width \(AD\) and \(BC\) be \(w_1\). For rectangle \(WXYZ\), let the length \(WX\) and \(YZ\) be \(l_2\) and the width \(WZ\) and \(XY\) be \(w_2\).
By counting the units on the grid (assuming each grid square has a side - length of 1 unit), for rectangle \(ABCD\), if \(AB = 4\) units and \(AD=3\) units. For rectangle \(WXYZ\), if \(WX = 2\) units and \(WZ = 6\) units.
Since \(l_1
eq l_2\) and \(w_1
eq w_2\) (even though \(l_1\times w_1=l_2\times w_2 = 12\) square units (if we assume a wrong initial count and correct it: if \(ABCD\) has length \(AB = 4\) and width \(AD = 6\) (counting the vertical and horizontal distances on the grid) and \(WXYZ\) has length \(WX= 2\) and width \(WZ = 12\)) ), a rigid transformation (translation, rotation, reflection) cannot change the side - lengths of a figure. Area being equal (\(A = l\times w\)) is not a sufficient condition for congruence. Also, just preserving right - angles (all rectangles have right - angles) is not a sufficient condition for congruence. A \(180^{\circ}\) rotation about the origin changes the position of a figure but not its shape and size, but since the side - lengths of \(ABCD\) and \(WXYZ\) are different, a \(180^{\circ}\) rotation (or any other rigid transformation) will not map \(ABCD\) to \(WXYZ\)

Answer:

No, there is not a rigid transformation or a combination of rigid transformations that will map \(ABCD\) to \(WXYZ\)