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 and size

First, note that figure PQRS is smaller than figure ABCD. So a dilation is needed. The scale factor can be found by comparing corresponding side lengths. Also, check the orientation.

Step2: Check the reflection

If we reflect figure ABCD across the y - axis, the orientation (in terms of left - right position relative to the y - axis) changes. Then, if we dilate with a scale factor. Let's assume a side length of ABCD is \(l_1\) and of PQRS is \(l_2\). If we take a horizontal side (for simplicity), say from a vertex of ABCD to another. Suppose a side of ABCD has length 6 units (by counting grid units) and a corresponding side of PQRS has length 2 units. The scale factor \(k=\frac{2}{6}=\frac{1}{3}\).

Step3: Eliminate other options

• Option B: Translation 9 units right changes the position but not the orientation in a way that would align with the similarity (the orientation of angles etc. is more consistent with a reflection - dilation).
• Option C: A \(90^{\circ}\) clockwise rotation changes the angle orientation in a way that does not match the given figures.
• Option D: A \(90^{\circ}\) counter - clockwise rotation changes the angle orientation in a way that does not match the given figures.

Answer:

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