Question

trapezoid ( fghi ) is the image of trapezoid fghi under a rotation ( 90^{circ} ) counterclockwise about the origin followed by a translation.
write the rules for the rotation and translation.
rotation: ( (x,y)\to(square,square) )
translation: ( (x,y)\to(square,square) )

Explanation:

Step1: Recall the rotation rule

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\).

Step2: Apply the rotation rule to a point

Take point \(G(-6,4)\). After rotation \((x = - 6,y = 4)\), using the rule \((x,y)\to(-y,x)\), we get \((-4,-6)\).

Step3: Determine the translation rule

The rotated point (after \(90^{\circ}\) counter - clockwise rotation of \(G(-6,4)\) is \((-4,-6)\), and the final image \(G'(2,-3)\).
To find the translation \((x,y)\to(x + a,y + b)\), we set up the equations:
For \(x\) - coordinate: \(-4+a=2\Rightarrow a = 6\)
For \(y\) - coordinate: \(-6 + b=-3\Rightarrow b=3\)

Answer:

Rotation: \((x,y)\to(-y,x)\)
Translation: \((x,y)\to(x + 6,y+3)\)