Question

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

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

For rectangle \(ABCD\), assume the side - lengths (by counting the grid units). Let's say \(AB = 4\) units and \(BC=6\) units. For rectangle \(WXYZ\), assume the side - lengths: let's say \(WX = 2\) units and \(XY = 12\) units.

Step3: Check the effect of rigid transformations

A rigid transformation (translation, rotation, reflection) cannot change the side - lengths of a figure. Since the side - lengths of \(ABCD\) (\(4\) and \(6\)) and \(WXYZ\) (\(2\) and \(12\)) are different, a rigid transformation or a combination of rigid transformations cannot map \(ABCD\) to \(WXYZ\).

• The first option is wrong because just having right angles (same shape in terms of angle measures) is not enough for congruence (size also matters).
• The second option is wrong because having the same area (\(A = l\times w\), \(4\times6=24\) and \(2\times12 = 24\)) but different side - lengths means the figures are not congruent (congruent figures must have equal corresponding side - lengths).
• The third option is wrong because a \(180^{\circ}\) rotation (a rigid transformation) preserves shape and size. But since the side - lengths are different, a \(180^{\circ}\) rotation 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 \(WXYZ\).