Question

kite ( ghij ) is the image of kite ( ghij ) under a rotation about the origin followed by a translation right 3 units and up 5 units. write the rules for the rotation and translation. rotation: ( (x,y)\to(square,square) ) translation: ( (x,y)\to(square,square) )

Explanation:

Step1: Find the coordinates of points before and after rotation (ignoring translation first)

Let's assume after rotation (before translation), we reverse the translation. For point \(I(-2,4)\), after translation \(I'\) is \((5,1)\). Reverse - translation: \(x = 5 - 3=2\), \(y = 1 - 5=-4\).
For a rotation of \(180^{\circ}\) about the origin, the rule is \((x,y)\to(-x,-y)\). If we take a general point \((x,y)\) and rotate \(180^{\circ}\) about the origin, we get \((-x,-y)\).

Step2: Determine the translation rule

The translation is right \(3\) units and up \(5\) units. The rule for translation is \((x,y)\to(x + 3,y + 5)\)

Answer:

Rotation: \((x,y)\to(-x,-y)\)
Translation: \((x,y)\to(x + 3,y + 5)\)