Question

rotate the figure 90° around (-1, 2). k(-5, 5) l(-2, 4) m(-4, 1)

Explanation:

Step1: Translate points

First, translate the rotation - center and the points so that the rotation - center is at the origin. Let the rotation - center $C(-1,2)$. For a point $P(x,y)$, the translated point $P'(x',y')$ is given by $x'=x + 1$ and $y'=y - 2$.
For $K(-5,5)$: $x_K'=-5 + 1=-4$, $y_K'=5 - 2 = 3$.
For $L(-2,4)$: $x_L'=-2 + 1=-1$, $y_L'=4 - 2 = 2$.
For $M(-4,1)$: $x_M'=-4 + 1=-3$, $y_M'=1 - 2=-1$.

Step2: Apply 90 - degree rotation formula

The formula for a 90 - degree counter - clockwise rotation about the origin is $(x,y)\to(-y,x)$.
For $K'(-4,3)$ after rotation: $x_{K''}=-3$, $y_{K''}=-4$.
For $L'(-1,2)$ after rotation: $x_{L''}=-2$, $y_{L''}=-1$.
For $M'(-3,-1)$ after rotation: $x_{M''}=1$, $y_{M''}=-3$.

Step3: Translate back

To get the final rotated points, reverse the first translation. Let the final rotated point be $P''(x_1,y_1)$. Then $x_1=x''-1$ and $y_1=y'' + 2$.
For $K''(-3,-4)$: $x_{K_f}=-3-1=-4$, $y_{K_f}=-4 + 2=-2$.
For $L''(-2,-1)$: $x_{L_f}=-2-1=-3$, $y_{L_f}=-1 + 2 = 1$.
For $M''(1,-3)$: $x_{M_f}=1-1 = 0$, $y_{M_f}=-3 + 2=-1$.

Answer:

$K'(-4,-2)$
$L'(-3,1)$
$M'(0,-1)$