Question

identify the transformation that maps the figure with center (7, 1) onto itself
a rotate 180° clockwise about (4, 1) and reflect across the x - axis
b rotate 180° clockwise about (4, 1) and reflect across the y - axis
c rotate 270° clockwise about (7, 1) and reflect across the x = 4
d rotate 270° clockwise about (7, 1) and reflect across the line x = 7

Explanation:

Step1: Analyze rotation and reflection properties

A rotation of \(270^{\circ}\) clockwise about the center \((7,1)\) followed by a reflection across the line \(x = 7\) (which is a vertical line passing through the center of the figure). For a figure with center \((7,1)\), rotating \(270^{\circ}\) clockwise about \((7,1)\) and then reflecting across \(x=7\) (a line of symmetry for the figure) will map the figure onto itself.

Step2: Check other options

• Option A: Rotating about \((4,1)\) (not the center of the figure \((7,1)\)) and reflecting across \(x -\) axis won't map the figure onto itself as the center of rotation is wrong.
• Option B: Rotating about \((4,1)\) (not the center of the figure \((7,1)\)) and reflecting across \(y -\) axis won't map the figure onto itself as the center of rotation is wrong.
• Option C: Reflecting across \(x = 4\) (not a line of symmetry for the figure with center \((7,1)\)) after rotation won't map the figure onto itself.

Answer:

D. rotate \(270^{\circ}\) clockwise about \((7,1)\) and reflect across the line \(x = 7\)