Question

use the drop - down menus to explain if the two figures below are congruent, similar, or neither. if the figures are similar, state the scale factor.
answer attempt 1 out of 2
figure ( h i j k ) congruent to figure ( v w x y ) because rigid motions be used to map figure ( h i j k ) onto figure ( v w x y ).
figure ( h i j k ) similar to figure ( v w x y ) because rigid motions and/or dilations be used to map figure ( h i j k ) onto figure ( v w x y ).

Explanation:

Step1: Check Congruence

Congruent figures have the same shape and size. Rigid motions (translations, rotations, reflections) preserve size. By counting grid - units, Figure \(HIJK\) has larger side lengths than Figure \(VWXY\). So, rigid motions cannot map \(HIJK\) onto \(VWXY\).

Step2: Check Similarity

Similar figures have the same shape. Let's assume \(HIJK\) and \(VWXY\) are quadrilaterals. For similarity, the ratios of corresponding side lengths should be equal.
Count the horizontal and vertical distances (using the grid).
Suppose for \(HIJK\), if we consider two adjacent sides: let's say the length of one side (horizontal) \(l_1 = 8\) units (counting grid - squares) and for \(VWXY\) the corresponding horizontal side \(l_2=4\) units. The ratio of side lengths \(\frac{l_2}{l_1}=\frac{4}{8}=\frac{1}{2}\).
If we check other corresponding sides (using the slope - concept for non - vertical/non - horizontal sides, but since the figures are on a grid, we can also count the 'diagonal' units in terms of right - triangle sides for each side of the quadrilaterals. For example, if a side of \(HIJK\) forms a right - triangle with legs \(a = 4\) and \(b = 4\) (length \(s_1=\sqrt{4^{2}+4^{2}} = 4\sqrt{2}\)) and a corresponding side of \(VWXY\) forms a right - triangle with legs \(a'=2\) and \(b' = 2\) (length \(s_2=\sqrt{2^{2}+2^{2}}=2\sqrt{2}\)), the ratio \(\frac{s_2}{s_1}=\frac{2\sqrt{2}}{4\sqrt{2}}=\frac{1}{2}\).
Since the ratios of all corresponding side lengths are equal (scale factor \(\frac{1}{2}\)), and we can use a dilation (scale factor \(\frac{1}{2}\)) along with rigid motions (translation, rotation, reflection) to map \(HIJK\) onto \(VWXY\).

Answer:

Figure \(HIJK\) is not congruent to Figure \(VWXY\) because rigid motions cannot be used to map Figure \(HIJK\) onto Figure \(VWXY\).
Figure \(HIJK\) is similar to Figure \(VWXY\) because rigid motions and/or dilations can be used to map Figure \(HIJK\) onto Figure \(VWXY\), and the scale factor is \(\frac{1}{2}\).