Question

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?

$r_{0,90^{circ}}$
$r_{0,180^{circ}}$
$r_{0,270^{circ}}$
$r_{0,360^{circ}}$

Explanation:

Step1: Recall rotation rules about the origin

For a $90^{\circ}$ counter - clockwise rotation about the origin $R_{0,90^{\circ}}$, the rule is $(x,y)\to(-y,x)$. For a $180^{\circ}$ rotation about the origin $R_{0,180^{\circ}}$, the rule is $(x,y)\to(-x,-y)$. For a $270^{\circ}$ counter - clockwise rotation about the origin $R_{0,270^{\circ}}$, the rule is $(x,y)\to(y, - x)$. For a $360^{\circ}$ rotation about the origin $R_{0,360^{\circ}}$, the rule is $(x,y)\to(x,y)$.

Step2: Test the points with the rules

For point $A(-3,4)$:

• For $R_{0,90^{\circ}}$: $(-3,4)\to(-4,-3)$ (not correct).
• For $R_{0,180^{\circ}}$: $(-3,4)\to(3,-4)$ (not correct).
• For $R_{0,270^{\circ}}$: $(-3,4)\to(4,3)$ (correct).
• For $R_{0,360^{\circ}}$: $(-3,4)\to(-3,4)$ (not correct).

For point $B(4,-5)$:

• For $R_{0,270^{\circ}}$: $(4,-5)\to(-5,-4)$ (correct).

For point $C(1,6)$:

• For $R_{0,270^{\circ}}$: $(1,6)\to(6,-1)$ (correct).

Answer:

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