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

For a point \((x,y)\) rotated about the origin \((0,0)\):

• \(R_{0,90^{\circ}}\): \((x,y)\to(-y,x)\)
• \(R_{0,180^{\circ}}\): \((x,y)\to(-x,-y)\)
• \(R_{0,270^{\circ}}\): \((x,y)\to(y,-x)\)
• \(R_{0,- 90^{\circ}}\) (equivalent to \(R_{0,270^{\circ}}\)): \((x,y)\to(y,-x)\)
• \(R_{0,-270^{\circ}}\) (equivalent to \(R_{0,90^{\circ}}\)): \((x,y)\to(-y,x)\)

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

• For \(R_{0,90^{\circ}}\): \((3,-2)\to(2,3)\) (since \(-y = 2\) and \(x = 3\))
• For \(R_{0,270^{\circ}}\): \((3,-2)\to(-2,-3)\)
• For \(R_{0,-90^{\circ}}\) (same as \(R_{0,270^{\circ}}\)): \((3,-2)\to(-2,-3)\)
• For \(R_{0,-270^{\circ}}\) (same as \(R_{0,90^{\circ}}\)): \((3,-2)\to(2,3)\)

Answer:

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