Question

4 △abc was rotated 90° counterclockwise around the origin to form △abc. △abc was dilated using a scale factor of 1/3 with a center of dilation at the origin. the result is △abc, shown in the coordinate plane. what are the coordinates of △abc? show your work. -2÷1/3 = -6 1÷1/3 =

Explanation:

Step1: Reverse the dilation

To reverse the dilation with a scale - factor of $\frac{1}{3}$, we multiply the coordinates of $\triangle A''B''C''$ by 3.
Let the coordinates of $A''( - 2,1)$, $B''(1,1)$, $C''(-2,-3)$. After reversing the dilation, the coordinates of $\triangle A'B'C'$ are:
For $A'$: $x_{A'}= - 2\times3=-6$, $y_{A'}=1\times3 = 3$;
For $B'$: $x_{B'}=1\times3 = 3$, $y_{B'}=1\times3 = 3$;
For $C'$: $x_{C'}=-2\times3=-6$, $y_{C'}=-3\times3=-9$.

Step2: Reverse the 90 - degree counter - clockwise rotation

The rule for a 90 - degree counter - clockwise rotation about the origin is $(x,y)\to(-y,x)$. To reverse it (a 90 - degree clockwise rotation), the rule is $(x,y)\to(y, - x)$.
For $A'(-6,3)$: After reversing the rotation, $x_{A}=3$, $y_{A}=6$.
For $B'(3,3)$: After reversing the rotation, $x_{B}=3$, $y_{B}=-3$.
For $C'(-6,-9)$: After reversing the rotation, $x_{C}=-9$, $y_{C}=6$.

Answer:

$A(3,6)$, $B(3, - 3)$, $C(-9,6)$