Question

which composition of similarity transformations maps polygon abcd to polygon abcd? a dilation with a scale factor of \\( \frac { 1 } { 4 } \\) and then a rotation a dilation with a scale factor of \\( \frac { 1 } { 4 } \\) and then a translation a dilation with a scale factor of 4 and then a rotation a dilation with a scale factor of 4 and then a translation

Explanation:

Step1: Determine the scale factor

Let's assume a side - length of polygon \(ABCD\). For example, if we consider a horizontal or vertical side (by counting the grid units). Suppose a side of \(ABCD\) has a length \(l\) (counting the number of units between two vertices). For the corresponding side of \(A'B'C'D'\), if the length of the side of \(ABCD\) is \(4\) units (by counting on the grid) and the length of the corresponding side of \(A'B'C'D'\) is \(1\) unit. The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of the image side}}{\text{length of the pre - image side}}\). So, \(k = \frac{1}{4}\).

Step2: Determine the transformation type

After dilation, the orientation of the polygon \(A'B'C'D'\) is different from that of \(ABCD\). A translation would just slide the figure without changing its orientation. A rotation changes the orientation of a figure. Since the orientation of the polygon has changed after the size change (dilation), the second transformation is a rotation.

Answer:

a dilation with a scale factor of \(\frac{1}{4}\) and then a rotation.