Question

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 origin followed by a dilation centered at the origin with scale factor \\( \frac { 1 } { 2 } \\)
\\( 90 ^ { \circ } \\) counterclockwise rotation around the origin followed by a dilation centered at the origin with scale factor \\( \frac { 1 } { 2 } \\)

Explanation:

Step1: Analyze each option

• Option 1: Dilation with scale factor \(\frac{1}{2}\) changes size, reflection across \(x -\)axis changes orientation. But not sure if it maps the figure.
• Option 2: Dilation with scale factor \(\frac{1}{2}\) changes size, translation 4 units up changes position. But not sure if it maps the figure.
• Option 3: \(90^{\circ}\) clock - wise rotation changes orientation, dilation with scale factor \(\frac{1}{2}\) changes size. But not sure if it maps the figure.
• Option 4: \(90^{\circ}\) counter - clockwise rotation around the origin followed by a dilation centered at the origin with scale factor \(\frac{1}{2}\).

Assume original figure has certain coordinates. After \(90^{\circ}\) counter - clockwise rotation \((x,y)\to(-y,x)\), then dilation \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\) which can map the figure.

Answer:

90° counterclockwise rotation around the origin followed by a dilation centered at the origin with scale factor \(\frac{1}{2}\)