Question

below to rotate the figure 270° counter - clockwise about the point (-1,4). fill out all of the corresponding points of the image figure. animate rotation again pre - image image (2,2) → (□,□) (4,3) → (□,□) (3,6) → (□,□) (2,5) → (□,□) try values

Explanation:

Step1: Translate the point of rotation to origin

To rotate a point $(x,y)$ counter - clockwise about a point $(a,b)$ by $270^{\circ}$, first translate the point of rotation $(a,b)$ to the origin. The translation rule is $(x',y')=(x - a,y - b)$. For a $270^{\circ}$ counter - clockwise rotation about the origin, the rotation rule is $(x_1,y_1)=(-y',x')$. Then reverse the translation: $(x_2,y_2)=(x_1 + a,y_1 + b)$.
Let $(a,b)=(-1,4)$.
For the point $(x,y)=(2,2)$:
First translation: $x'=2-(-1)=3$, $y'=2 - 4=-2$.

Step2: Rotate about the origin

For the $270^{\circ}$ counter - clockwise rotation about the origin, $x_1=-y'=2$, $y_1=x'=3$.

Step3: Reverse the translation

$x_2=2+(-1)=1$, $y_2=3 + 4=7$.

For the point $(x,y)=(4,3)$:
First translation: $x'=4-(-1)=5$, $y'=3 - 4=-1$.
Second step (rotation about origin): $x_1=-y'=1$, $y_1=x'=5$.
Third step (reverse translation): $x_2=1+(-1)=0$, $y_2=5 + 4=9$.

For the point $(x,y)=(3,6)$:
First translation: $x'=3-(-1)=4$, $y'=6 - 4=2$.
Second step (rotation about origin): $x_1=-y'=-2$, $y_1=x'=4$.
Third step (reverse translation): $x_2=-2+(-1)=-3$, $y_2=4 + 4=8$.

For the point $(x,y)=(2,5)$:
First translation: $x'=2-(-1)=3$, $y'=5 - 4=1$.
Second step (rotation about origin): $x_1=-y'=-1$, $y_1=x'=3$.
Third step (reverse translation): $x_2=-1+(-1)=-2$, $y_2=3 + 4=7$.

Answer:

$(2,2)\to(1,7)$
$(4,3)\to(0,9)$
$(3,6)\to(-3,8)$
$(2,5)\to(-2,7)$