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°}

□ r_{0,180°}

□ r_{0,270°}

□ r_{0,-90°}

□ r_{0,-270°}

Explanation:

Step1: Recall rotation rules

• For a \(90^{\circ}\) counter - clockwise rotation (\(R_{0,90^{\circ}}\)) about the origin, the rule is \((x,y)\to(-y,x)\). For the point \((3,-2)\), we get \((-(-2),3)=(2,3)\).
• For a \(270^{\circ}\) clockwise rotation (\(R_{0,270^{\circ}}\)), since a \(270^{\circ}\) clockwise rotation is equivalent to a \(90^{\circ}\) counter - clockwise rotation. The rule for a \(270^{\circ}\) clockwise rotation about the origin is also \((x,y)\to(-y,x)\). For the point \((3, - 2)\), we have \((-(-2),3)=(2,3)\).
• For a \(-90^{\circ}\) rotation (which is \(270^{\circ}\) counter - clockwise), the rule is \((x,y)\to(y, - x)\). For the point \((3,-2)\), we get \((-2,-3)\).
• For a \(180^{\circ}\) rotation, the rule is \((x,y)\to(-x,-y)\). For the point \((3,-2)\), we get \((-3,2)\).
• For a \(-270^{\circ}\) rotation (which is \(90^{\circ}\) clockwise), the rule is \((x,y)\to(y, - x)\). For the point \((3,-2)\), we get \((-2,-3)\).

Answer:

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