Question

graph the polygon with the given vertices and its image after the given rotation about point a. a(-2, -3), b(-3,2), c(3, -1); 180°

Explanation:

Step1: Recall rotation rule

A $180^{\circ}$ rotation about a point $(x_0,y_0)$ for a point $(x,y)$ is given by the rule $(x,y)\to(2x_0 - x,2y_0 - y)$. Here $x_0=-2,y_0 = - 3$.

Step2: Find new coordinates of point B

For $B(-3,2)$:
$x'=2\times(-2)-(-3)=-4 + 3=-1$
$y'=2\times(-3)-2=-6 - 2=-8$
So $B'(-1,-8)$.

Step3: Find new coordinates of point C

For $C(3,-1)$:
$x'=2\times(-2)-3=-4-3=-7$
$y'=2\times(-3)-(-1)=-6 + 1=-5$
So $C'(-7,-5)$.

Step4: Graph the polygons

First, plot the original polygon with vertices $A(-2,-3),B(-3,2),C(3,-1)$. Then plot the new polygon with vertices $A(-2,-3),B'(-1,-8),C'(-7,-5)$.

Answer:

The original polygon has vertices $A(-2,-3),B(-3,2),C(3,-1)$ and the rotated polygon (after $180^{\circ}$ rotation about $A$) has vertices $A(-2,-3),B'(-1,-8),C'(-7,-5)$.