Question

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

Explanation:

Step1: Identify original coordinates

From the graph, $S=(0, - 3)$, $T=(0,9)$, $U=(3,9)$, $V=(3,-9)$.

Step2: Apply dilation formula

For a dilation centered at the origin with scale - factor $k=\frac{1}{3}$, if a point has coordinates $(x,y)$, its new coordinates $(x',y')$ after dilation are given by $(x',y')=(k x,k y)$.
For point $S=(0, - 3)$:
$x'=\frac{1}{3}\times0 = 0$, $y'=\frac{1}{3}\times(-3)=-1$, so $S'=(0,-1)$.
For point $T=(0,9)$:
$x'=\frac{1}{3}\times0 = 0$, $y'=\frac{1}{3}\times9 = 3$, so $T'=(0,3)$.
For point $U=(3,9)$:
$x'=\frac{1}{3}\times3 = 1$, $y'=\frac{1}{3}\times9 = 3$, so $U'=(1,3)$.
For point $V=(3,-9)$:
$x'=\frac{1}{3}\times3 = 1$, $y'=\frac{1}{3}\times(-9)=-3$, so $V'=(1,-3)$.

Answer:

$S'(0,-1)$
$T'(0,3)$
$U'(1,3)$
$V'(1,-3)$