Question

△abc was rotated 90° counter - clockwise around the origin to form △abc. △abc was dilated using a scale factor of $\frac{1}{3}$ 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.

Explanation:

Step1: Reverse the dilation

If a point $(x,y)$ is dilated by a scale factor $k$ with the center of dilation at the origin to get $(x',y')$, then the original - point before dilation is given by $(x/k,y/k)$. Here $k = \frac{1}{3}$, so to reverse the dilation, we multiply the coordinates of the points of $\triangle A''B''C''$ by 3.

Step2: Reverse the rotation

The rule for a $90^{\circ}$ counter - clockwise rotation around the origin is $(x,y)\to(-y,x)$. To reverse a $90^{\circ}$ counter - clockwise rotation, we use the rule $(x,y)\to(y, - x)$.

Let's assume the coordinates of $A''$, $B''$, and $C''$ from the graph are $A''(-1,1)$, $B''(1,1)$, $C''(-2,-3)$.

Reverse the dilation:

For point $A''(-1,1)$: After reversing the dilation (multiplying by 3), we get $A'(-3,3)$.
For point $B''(1,1)$: After reversing the dilation (multiplying by 3), we get $B'(3,3)$.
For point $C''(-2,-3)$: After reversing the dilation (multiplying by 3), we get $C'(-6,-9)$.

Reverse the rotation:

For point $A'(-3,3)$: After reversing the $90^{\circ}$ counter - clockwise rotation, we get $A(3,3)$.
For point $B'(3,3)$: After reversing the $90^{\circ}$ counter - clockwise rotation, we get $B(3, - 3)$.
For point $C'(-6,-9)$: After reversing the $90^{\circ}$ counter - clockwise rotation, we get $C(-9,6)$.

Answer:

The coordinates of $\triangle ABC$ are $A(3,3)$, $B(3, - 3)$, $C(-9,6)$