Question

in the diagram, figure abcd ~ figure pqrs. which sequence of rigid motions or dilations, or both, shows this similarity? a reflection across the y - axis followed by a dilation centered at the origin with scale factor 1/3 b translation 9 units right followed by a dilation centered at the origin with scale factor 1/3 c dilation centered at the origin with scale factor 1/3 followed by a 90° clockwise rotation around the origin d dilation centered at the origin with scale factor 1/3 followed by a 90° counterclockwise rotation around the origin

Explanation:

Step1: Analyze the orientation

First, observe the orientation of the two figures. Figure \(ABCD\) and figure \(PQRS\) are mirror - images with respect to the \(y\) - axis. A reflection across the \(y\) - axis changes the \(x\) - coordinate of a point \((x,y)\) to \((-x,y)\).

Step2: Analyze the size

Next, consider the size of the figures. Let's assume a side length of figure \(ABCD\). If we count the grid units for a side of \(ABCD\) (say \(AB\)) and compare it to the corresponding side of \(PQRS\) (say \(PQ\)). Suppose the length of a side of \(ABCD\) is \(6\) units (by counting grid - squares between two vertices) and the length of the corresponding side of \(PQRS\) is \(2\) units. The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\). Here, \(k = \frac{1}{3}\)

A translation (Option B) would just shift the figure without changing its orientation in the way shown. A rotation (Options C and D) would change the angle of the figure's sides with respect to the axes in a non - mirror - image way. A reflection across the \(y\) - axis (which changes the left - right orientation) followed by a dilation (which changes the size) is the correct transformation.

Answer:

A. Reflection across the y - axis followed by a dilation centered at the origin with scale factor \( \frac{1}{3}\)