Question

rotate the figure 270° around (-5, -4). k(-13, -3) l(-7, 3) m(-6, -2)

Explanation:

Step1: Translation rule

First, translate the rotation - center and points so that the rotation - center is at the origin. Let the rotation - center $C(-5,-4)$. For a point $P(x,y)$, the translated point $P'(x - x_C,y - y_C)$.
For point $K(-13,-3)$: $x_{K'}= - 13-(-5)=-8$, $y_{K'}= - 3-(-4)=1$.
For point $L(-7,3)$: $x_{L'}= - 7-(-5)=-2$, $y_{L'}=3 - (-4)=7$.
For point $M(-6,-2)$: $x_{M'}= - 6-(-5)=-1$, $y_{M'}= - 2-(-4)=2$.

Step2: 270 - degree rotation rule about the origin

The rule for rotating a point $(x,y)$ 270 degrees counter - clockwise about the origin is $(x,y)\to(y, - x)$.
For the translated point $K'(-8,1)$: After rotation, $x_{K''}=1$, $y_{K''}=8$.
For the translated point $L'(-2,7)$: After rotation, $x_{L''}=7$, $y_{L''}=2$.
For the translated point $M'(-1,2)$: After rotation, $x_{M''}=2$, $y_{M''}=1$.

Step3: Reverse - translation

To get the final rotated points, we reverse the translation. Let the final rotated points be $K^*(x,y)$, $L^*(x,y)$, $M^*(x,y)$. The formula is $x=x_{K''}+x_C$, $y=y_{K''}+y_C$ (similarly for $L$ and $M$).
For the rotated - and then reversed - translated point $K$: $x = 1+(-5)=-4$, $y = 8+(-4)=4$.
For the rotated - and then reversed - translated point $L$: $x = 7+(-5)=2$, $y = 2+(-4)=-2$.
For the rotated - and then reversed - translated point $M$: $x = 2+(-5)=-3$, $y = 1+(-4)=-3$.

Answer:

$K'(-4,4)$
$L'(2,-2)$
$M'(-3,-3)$