Question

josefina wants to prove that figures d and d are similar.
what sequence of transformations could josefina perform to prove that the figures are similar?
josefina could translate figure d left 3 units to form figure d.
then she could dilate figure d by a scale factor of 2 with a center of dilation at (0,0) to form figure d

Explanation:

Step1: Analyze the translation

Looking at the horizontal distance between corresponding points of Figure \(D\) and Figure \(D'\). If we assume a point on \(D\) and its corresponding point on \(D'\), we can count the units. By visual inspection (assuming a grid - based coordinate system), we find that Figure \(D\) is translated \(3\) units up to form Figure \(D'\).

Step2: Analyze the dilation

For the dilation of Figure \(D'\) to form \(D''\). The scale factor \(k\) can be found by comparing the lengths of corresponding sides. If we assume a side length of \(D'\) is \(l_1\) and the corresponding side length of \(D''\) is \(l_2\). By visual inspection (using the grid), if \(l_1 = 2\) units and \(l_2=4\) units, then the scale factor \(k=\frac{l_2}{l_1}=2\). The center of dilation is \((0,0)\) (since the figure is expanded from the origin).

Answer:

Translate Figure \(D\) \(3\) units up to form Figure \(D'\), then dilate Figure \(D'\) by a scale factor of \(2\) with a center of dilation at \((0,0)\) to form Figure \(D''\)