Question

a path between 3 cities is drawn on a map, creating the outlined region shown in the graph below. which sequence of transformations would result in mapping the region onto itself? a. a rotation 90 - degrees counterclockwise about the origin and then a rotation 270 - degrees counterclockwise about the origin b. a rotation 180 - degrees counterclockwise about the origin and then a rotation 270 - degrees counterclockwise about the origin c. a rotation 90 - degrees counterclockwise about the origin and then a rotation 180 - degrees counterclockwise about the origin d. a reflection across the y - axis and then a reflection across the x - axis

Explanation:

Step1: Analyze option A

A 90 - degree counter - clockwise rotation followed by a 270 - degree counter - clockwise rotation about the origin. The sum of the rotation angles is \(90 + 270=360\) degrees. A 360 - degree rotation about the origin maps a figure onto itself.

Step2: Analyze option B

A 180 - degree counter - clockwise rotation followed by a 270 - degree counter - clockwise rotation about the origin. The sum of the rotation angles is \(180+270 = 450\) degrees. \(450\div360 = 1\cdots\cdots90\), so it is equivalent to a 90 - degree rotation, and the figure will not map onto itself.

Step3: Analyze option C

A 90 - degree counter - clockwise rotation followed by a 180 - degree counter - clockwise rotation about the origin. The sum of the rotation angles is \(90 + 180=270\) degrees. The figure will not map onto itself.

Step4: Analyze option D

A reflection across the \(y\) - axis and then a reflection across the \(x\) - axis is equivalent to a 180 - degree rotation about the origin. The figure will not map onto itself.

Answer:

A. a rotation 90 - degrees counterclockwise about the origin and then a rotation 270 - degrees counterclockwise about the origin