Question

one vertex of a polygon is located at (3, -2). after a rotation, the vertex is located at (2, 3). which transformations could have taken place? select two options. consider both clockwise and counterclockwise rotations when answering this question. $r_{0,90^{circ}}$ $r_{0,180^{circ}}$ $r_{0,270^{circ}}$ $r_{0,-90^{circ}}$ $r_{0,-270^{circ}}$

Explanation:

Step1: Recall rotation rules

The general rules for rotation about the origin \((0,0)\) are:
For a \(90^{\circ}\) counter - clockwise rotation \(R_{0,90^{\circ}}\) of a point \((x,y)\), the new point is \((-y,x)\).
For a \(180^{\circ}\) counter - clockwise rotation \(R_{0,180^{\circ}}\) of a point \((x,y)\), the new point is \((-x,-y)\).
For a \(270^{\circ}\) counter - clockwise rotation \(R_{0,270^{\circ}}\) of a point \((x,y)\), the new point is \((y, - x)\).
For a \(90^{\circ}\) clockwise rotation \(R_{0,-90^{\circ}}\) of a point \((x,y)\), the new point is \((y,-x)\).
For a \(270^{\circ}\) clockwise rotation \(R_{0,-270^{\circ}}\) of a point \((x,y)\), the new point is \((-y,x)\).

Step2: Apply rules to the point \((3,-2)\)

For \(R_{0,90^{\circ}}\): Given \((x = 3,y=-2)\), the new point is \((-(-2),3)=(2,3)\).
For \(R_{0,-270^{\circ}}\) (which is equivalent to \(R_{0,90^{\circ}}\)), given \((x = 3,y=-2)\), the new point is \((-(-2),3)=(2,3)\).

Answer:

\(R_{0,90^{\circ}},R_{0,-270^{\circ}}\)