Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{3}$, centered at the origin.

Explanation:

Step1: Find the original coordinates

From the graph, the coordinates of point $B$ are $(- 9,9)$, the coordinates of point $C$ are $(9,9)$, and the coordinates of point $D$ are $(-9,-9)$.

Step2: Apply the dilation formula

The formula for dilation centered at the origin with a scale - factor $k$ is $(x,y)\to(kx,ky)$. Here $k = \frac{1}{3}$.
For point $B(-9,9)$:
$x=-9,y = 9,k=\frac{1}{3}$, then the new coordinates are $(\frac{1}{3}\times(-9),\frac{1}{3}\times9)=(-3,3)$.
For point $C(9,9)$:
$x = 9,y = 9,k=\frac{1}{3}$, then the new coordinates are $(\frac{1}{3}\times9,\frac{1}{3}\times9)=(3,3)$.
For point $D(-9,-9)$:
$x=-9,y=-9,k=\frac{1}{3}$, then the new coordinates are $(\frac{1}{3}\times(-9),\frac{1}{3}\times(-9))=(-3,-3)$.

Answer:

The coordinates of $B$ after dilation are $(-3,3)$, the coordinates of $C$ after dilation are $(3,3)$, and the coordinates of $D$ after dilation are $(-3,-3)$.