Question

writing a rule for a rotation
after a rotation a(-3, 4) maps to a(4, 3), b(4, -5) maps to b(-5, -4), and c(1, 6) maps to c(6, -1). which rule describes the rotation?
$r_{0,180^{circ}}$ $r_{0,360^{circ}}$ $r_{0,90^{circ}}$ $r_{0,270^{circ}}$

Explanation:

Step1: Recall rotation rules

For a rotation of $90^{\circ}$ counter - clockwise about the origin $(x,y)\to(-y,x)$. For a rotation of $180^{\circ}$ about the origin $(x,y)\to(-x,-y)$. For a rotation of $270^{\circ}$ counter - clockwise about the origin $(x,y)\to(y, - x)$. For a rotation of $360^{\circ}$ about the origin $(x,y)\to(x,y)$.

Step2: Analyze point A

Given $A(-3,4)$ maps to $A'(4,3)$. If we consider the rule for a $270^{\circ}$ counter - clockwise rotation about the origin, for a point $(x,y)=(-3,4)$, applying the rule $(x,y)\to(y,-x)$ gives $(4,3)$.

Step3: Analyze point B

Given $B(4, - 5)$ maps to $B'(-5,-4)$. For the point $(x,y)=(4,-5)$, applying the $270^{\circ}$ counter - clockwise rotation rule $(x,y)\to(y,-x)$ gives $(-5,-4)$.

Step4: Analyze point C

Given $C(1,6)$ maps to $C'(6,-1)$. For the point $(x,y)=(1,6)$, applying the $270^{\circ}$ counter - clockwise rotation rule $(x,y)\to(y,-x)$ gives $(6,-1)$.

Answer:

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