Question

use the drop - down menus to explain if the two figures below are congruent, similar, or neither. if the figures are simi state the scale factor.
answer
figure uvwx congruent to figure defg because rigid motions be used to map figure uvwx onto figure defg.
figure uvwx similar to figure defg because rigid motions and/or dilations be used to map figure uvwx onto figure defg.

Explanation:

Step1: Understand congruent figures

Congruent figures have the same shape and size. Rigid motions (translations, rotations, reflections) preserve shape and size. If we check the side - lengths of the two quadrilaterals \(UVWX\) and \(DEFG\) using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For example, if we assume coordinates: Let \(X(0,6)\), \(W(5,7)\), \(V(8,5)\), \(U(6,5)\) and \(D(- 6,-6)\), \(E(-3,-8)\), \(F(-2,-2)\), \(G(-5,-3)\).
The length of \(XW=\sqrt{(5 - 0)^2+(7 - 6)^2}=\sqrt{25 + 1}=\sqrt{26}\), and the length of \(DE=\sqrt{(-3+6)^2+(-8 + 6)^2}=\sqrt{9+4}=\sqrt{13}\). Since the side - lengths are not equal, rigid motions (which preserve length) cannot map \(UVWX\) to \(DEFG\). So, Figure \(UVWX\) is not congruent to Figure \(DEFG\) because rigid motions cannot be used to map Figure \(UVWX\) onto Figure \(DEFG\).

Step2: Understand similar figures

Similar figures have the same shape (corresponding angles are equal) and the ratios of their corresponding side - lengths are equal (scale factor).
Let's calculate the ratios of side - lengths. If we assume two corresponding sides \(s_1\) (from \(UVWX\)) and \(s_2\) (from \(DEFG\)).
Since the side - lengths of \(UVWX\) and \(DEFG\) do not have a constant ratio (for example, if we consider the ratio of \(XW\) to \(DE\) as calculated above \(\frac{\sqrt{26}}{\sqrt{13}}=\sqrt{2}\), and if we check another pair of sides, say \(UV\) (length \(2\)) and \(DG\) (length \(\sqrt{(-5 + 6)^2+(-3 + 6)^2}=\sqrt{1 + 9}=\sqrt{10}\), \(\frac{2}{\sqrt{10}}=\frac{\sqrt{10}}{5}
eq\sqrt{2}\)), rigid motions and dilations (which change size by a constant scale factor) cannot be used to map \(UVWX\) to \(DEFG\).

Answer:

Figure \(UVWX\) is not congruent to Figure \(DEFG\) because rigid motions cannot be used to map Figure \(UVWX\) onto Figure \(DEFG\). Figure \(UVWX\) is not similar to Figure \(DEFG\) because rigid motions and/or dilations cannot be used to map Figure \(UVWX\) onto Figure \(DEFG\).