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. r0,90° r0,180° r0,270° r0,-90° r0,-270°

Explanation:

Step1: Recall rotation rules

The general rule for a rotation of $90^{\circ}$ counter - clockwise ($R_{0,90^{\circ}}$) about the origin is $(x,y)\to(-y,x)$, for a $180^{\circ}$ rotation ($R_{0,180^{\circ}}$) is $(x,y)\to(-x,-y)$, for a $270^{\circ}$ counter - clockwise ($R_{0,270^{\circ}}$) or $90^{\circ}$ clockwise ($R_{0, - 90^{\circ}}$) is $(x,y)\to(y,-x)$, and for a $270^{\circ}$ clockwise ($R_{0,-270^{\circ}}$) or $90^{\circ}$ counter - clockwise ($R_{0,90^{\circ}}$) is $(x,y)\to(-y,x)$.

Step2: Apply rules to the point

For the point $(3,-2)$:

• For $R_{0,90^{\circ}}$: $(3,-2)\to(2,3)$ (since $x = 3,y=-2$ and using the rule $(x,y)\to(-y,x)$ we get $-(-2)=2$ and $x = 3$).
• For $R_{0,- 90^{\circ}}$: $(3,-2)\to(-2,-3)$ (using the rule $(x,y)\to(y,-x)$: $y=-2$ and $-x=-3$).
• For $R_{0,180^{\circ}}$: $(3,-2)\to(-3,2)$.
• For $R_{0,270^{\circ}}$: $(3,-2)\to(-2,3)$ (since it is equivalent to a $90^{\circ}$ clockwise rotation and using the rule $(x,y)\to(y,-x)$ with $x = 3,y=-2$ gives $y=-2$ and $-x=-3$).
• For $R_{0,-270^{\circ}}$: $(3,-2)\to(2,3)$ (equivalent to a $90^{\circ}$ counter - clockwise rotation).

Answer:

$R_{0,90^{\circ}}$, $R_{0,-270^{\circ}}$