Question

rectangle ( cdef ) is the image of rectangle ( cdef ) under a rotation about the origin followed by a translation. write the rules for the rotation and translation. rotation: ( (x,y)mapsto(square,square) ) translation: ( (x,y)mapsto(square,square) )

Explanation:

Step1: Determine the rotation rule

• First, find the coordinates of a point before rotation. Let's take point \(C(4,7)\).
• If we rotate a point \((x,y)\) \(90^{\circ}\) clockwise about the origin, the rule is \((x,y)\to(y, - x)\). For \(C(4,7)\), after \(90^{\circ}\) clockwise rotation, it becomes \((7,-4)\).

Step2: Determine the translation rule

• After rotation, assume the point is \((x_1,y_1)\). Let's consider the relationship between the rotated - point (from step 1) and the final image point.
• Let's take the rotated - point (from step 1) \((7,-4)\) (corresponding to \(C'\)). The actual \(C'(-3,-2)\).
• We use the translation rule \((x,y)\to(x + a,y + b)\). For \(x\) - coordinate: \(-3=7+a\), so \(a=-10\). For \(y\) - coordinate: \(-2=-4 + b\), so \(b = 2\).

Answer:

Rotation: \((x,y)\to(y,-x)\)
Translation: \((x,y)\to(x - 10,y + 2)\)