Question

3.3b hw honors
rotate the figure 270° around (-1, 3). a(2, 1) b(5, 1) c(8, -4)
a(

)
b(

)
c(

)

Explanation:

Step1: Translate points relative to center

To rotate a point $(x,y)$ 270 - degrees around a center $(a,b)$, first translate the point so that the center is at the origin. Let $(x,y)$ be the original point and $(a,b)$ be the center of rotation. The translated - point $(x_1,y_1)$ is given by $x_1=x - a$ and $y_1=y - b$.
For point $A(2,1)$ with center $(-1,3)$: $x_1=2-(-1)=3$, $y_1=1 - 3=-2$.
For point $B(5,1)$ with center $(-1,3)$: $x_1=5-(-1)=6$, $y_1=1 - 3=-2$.
For point $C(8,-4)$ with center $(-1,3)$: $x_1=8-(-1)=9$, $y_1=-4 - 3=-7$.

Step2: Apply 270 - degree rotation rule

The rule for a 270 - degree counter - clockwise rotation about the origin is $(x,y)\to(y,-x)$.
For the translated point of $A$: $(x_1,y_1)=(3,-2)\to(-2,-3)$.
For the translated point of $B$: $(x_1,y_1)=(6,-2)\to(-2,-6)$.
For the translated point of $C$: $(x_1,y_1)=(9,-7)\to(-7,-9)$.

Step3: Translate back

To get the final rotated points, we translate back using the rule $x_2=x_1 + a$ and $y_2=y_1 + b$.
For the rotated point of $A$: $x_2=-2+(-1)=-3$, $y_2=-3 + 3=0$.
For the rotated point of $B$: $x_2=-2+(-1)=-3$, $y_2=-6 + 3=-3$.
For the rotated point of $C$: $x_2=-7+(-1)=-8$, $y_2=-9 + 3=-6$.

Answer:

$A'(-3,0)$
$B'(-3,-3)$
$C'(-8,-6)$