Question

what transformations were applied to abcd to obtain abcd? a. rotate 180 degrees counterclockwise about the origin, then translate 2 units left. b. rotate 90 degrees counterclockwise about the origin, then translate 2 units left. c. rotate 90 degrees counterclockwise about the origin, then

Explanation:

Step1: Analyze rotation

If we rotate a point $(x,y)$ 180 - degrees counter - clockwise about the origin, the transformation rule is $(x,y)\to(-x,-y)$. If we rotate a point $(x,y)$ 90 - degrees counter - clockwise about the origin, the transformation rule is $(x,y)\to(-y,x)$. By observing the orientation of the rectangle $ABCD$ and $A'B'C'D'$, we can see that the orientation has changed such that if we consider a vertex of $ABCD$, say $A(1,4)$, to get to the corresponding vertex $A'(- 4, - 1)$ of $A'B'C'D'$, we first rotate 180 degrees counter - clockwise about the origin.

Step2: Analyze translation

After rotation, we note the horizontal shift. The $x$ - coordinates of the corresponding points of the rotated figure and $A'B'C'D'$ show a left - ward shift. For example, after rotating a point and then comparing its position to the final position in $A'B'C'D'$, we find that there is a 2 - unit left translation.

Answer:

A. Rotate 180 degrees counterclockwise about the origin, then translate 2 units left.