Question

in the diagram, figure ( a b c d sim ) figure ( p q r s ).
which sequence of rigid motions or dilations, or both, shows this similarity?
dilation centered at the origin with scale factor ( \frac{1}{2} ) followed by a reflection across the ( x )-axis
dilation centered at the origin with scale factor ( \frac{1}{2} ) followed by a translation 4 units up
( 90^{circ} ) clockwise rotation around the orgin followed by a dilaton centered at the origin with scale factor ( \frac{1}{2} )
( 90^{circ} ) counterclockwise rotation around the origin followed by a dilaton centered at the origin with scale factor ( \frac{1}{2} )

Explanation:

Step1: Analyze the size relationship

First, observe the size of the two figures. The side - length of figure \(PQRS\) is half of the side - length of figure \(ABCD\). So, a dilation with a scale factor of \(\frac{1}{2}\) is involved.

Step2: Analyze the orientation relationship

Then, consider the orientation. If we first perform a dilation of figure \(ABCD\) centered at the origin with a scale factor of \(\frac{1}{2}\), and then reflect the dilated figure across the \(x\) - axis, we can get figure \(PQRS\).
Let's check other options:

• For option B: If we do a translation 4 units up after dilation, the orientation (the relative position of the vertices with respect to the \(x\) and \(y\) axes) will not match.
• For option C: A \(90^{\circ}\) clockwise rotation around the origin followed by a dilation will not give the correct orientation and position relationship.
• For option D: A \(90^{\circ}\) counter - clockwise rotation around the origin followed by a dilation will not give the correct orientation and position relationship.

Answer:

Dilation centered at the origin with scale factor \(\frac{1}{2}\) followed by a reflection across the \(x\) - axis.