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: Analyze the scale factor

The size of polygon \(A'B'C'D'\) is smaller than polygon \(ABCD\). The scale factor \(k\) of a dilation is given by the ratio of the side - lengths of the image to the pre - image. If we assume a side - length of \(ABCD\) (for example, if we consider a horizontal or vertical segment that we can count the units on the grid). Let's say a side of \(ABCD\) has a length of \(4\) units (by counting grid squares) and the corresponding side of \(A'B'C'D'\) has a length of \(1\) unit. The scale factor \(k=\frac{\text{length of image side}}{\text{length of pre - image side}}=\frac{1}{4}\).

Step2: Analyze the transformation type (rotation vs translation)

A translation is a slide, which moves every point of a figure or space by the same distance in a given direction. A rotation is a transformation that turns a figure around a fixed point. If we consider the orientation of the polygons: Polygon \(A'B'C'D'\) has a different orientation than polygon \(ABCD\). For example, if we look at the slope of the sides. If we assume a reference side (say \(AB\) in \(ABCD\) and \(A'B'\) in \(A'B'C'D'\)), the direction (angle) of the sides has changed. A translation would not change the orientation of the figure, while a rotation changes the orientation of the figure.

Answer:

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