Question

rotating a figure
which shows the image of rectangle abcd after the rotation (x, y) → (-y, x)?

Explanation:

Step1: Find coordinates of rectangle ABCD

Assume \(A=(1,0)\), \(B=(2,0)\), \(C=(2,4)\), \(D=(1,4)\)

Step2: Apply the rotation rule \((x,y)\to(-y,x)\)

For point \(A(1,0)\): \(x = 1,y = 0\), after rotation \((-0,1)=(0,1)\)
For point \(B(2,0)\): \(x = 2,y = 0\), after rotation \((-0,2)=(0,2)\)
For point \(C(2,4)\): \(x = 2,y = 4\), after rotation \((-4,2)\)
For point \(D(1,4)\): \(x = 1,y = 4\), after rotation \((-4,1)\)

Step3: Analyze the position of the rotated rectangle

The original rectangle is vertical on the left - side of \(y\) - axis. After a \(90^{\circ}\) counter - clockwise rotation (since the rule \((x,y)\to(-y,x)\) is equivalent to a \(90^{\circ}\) counter - clockwise rotation about the origin), the new rectangle should be horizontal below the \(x\) - axis.

Answer:

The third option (the one where the rotated rectangle \(A'B'C'D'\) is horizontal below the \(x\) - axis)