Question

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

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 $(x - a,y - b)$ is considered.
For point A(2,1) with center (-1,3), the translated point is $(2-(-1),1 - 3)=(3,-2)$.
For point B(5,1) with center (-1,3), the translated point is $(5-(-1),1 - 3)=(6,-2)$.
For point C(8,-4) with center (-1,3), the translated point is $(8-(-1),-4 - 3)=(9,-7)$.

Step2: Apply 270 - degree rotation rule

The rule for a 270 - degree counter - clockwise rotation of a point $(x,y)$ around the origin is $(y,-x)$.
For the translated point of A(3,-2), the rotated point is $(-2,-3)$.
For the translated point of B(6,-2), the rotated point is $(-2,-6)$.
For the translated point of C(9,-7), the rotated point is $(-7,-9)$.

Step3: Translate back

To get the final rotated point, we translate the rotated - and - translated point back to the original coordinate system. The formula is $(x + a,y + b)$.
For the rotated point of A(-2,-3) with center (-1,3), the final point A' is $(-2+(-1),-3 + 3)=(-3,0)$.
For the rotated point of B(-2,-6) with center (-1,3), the final point B' is $(-2+(-1),-6 + 3)=(-3,-3)$.
For the rotated point of C(-7,-9) with center (-1,3), the final point C' is $(-7+(-1),-9 + 3)=(-8,-6)$.

Answer:

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