Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{4}$, centered at the origin.

Explanation:

Step1: Identify original coordinates

$P(4, - 8)$, $Q(4,4)$, $R(8,8)$, $S(8,-4)$

Step2: Apply dilation formula

For a dilation centered at the origin with scale - factor $k=\frac{1}{4}$, the new coordinates $(x',y')$ of a point $(x,y)$ are given by $(x',y')=(k x,k y)$.
For point $P(4, - 8)$:
$x'_P=\frac{1}{4}\times4 = 1$, $y'_P=\frac{1}{4}\times(-8)=-2$
For point $Q(4,4)$:
$x'_Q=\frac{1}{4}\times4 = 1$, $y'_Q=\frac{1}{4}\times4 = 1$
For point $R(8,8)$:
$x'_R=\frac{1}{4}\times8 = 2$, $y'_R=\frac{1}{4}\times8 = 2$
For point $S(8,-4)$:
$x'_S=\frac{1}{4}\times8 = 2$, $y'_S=\frac{1}{4}\times(-4)=-1$

Answer:

$P'(1,-2)$, $Q'(1,1)$, $R'(2,2)$, $S'(2,-1)$