Question

graph the image of rectangle rstu after a dilation with a scale factor of ⅓, centered at the origin.

Explanation:

Step1: Recall dilation formula

For a point $(x,y)$ dilated with a scale - factor $k$ centered at the origin, the new point $(x',y')$ is given by $(x',y')=(kx,ky)$. Here $k = \frac{1}{3}$.

Step2: Find coordinates of dilated points

The coordinates of rectangle $RSTU$ are: $R(-6,- 2)$, $S(8,-2)$, $T(8,6)$, $U(-6,6)$.
For point $R(-6,-2)$:
$x'_R=\frac{1}{3}\times(-6)=-2$, $y'_R=\frac{1}{3}\times(-2)=-\frac{2}{3}$.
For point $S(8,-2)$:
$x'_S=\frac{1}{3}\times8=\frac{8}{3}$, $y'_S=\frac{1}{3}\times(-2)=-\frac{2}{3}$.
For point $T(8,6)$:
$x'_T=\frac{1}{3}\times8=\frac{8}{3}$, $y'_T=\frac{1}{3}\times6 = 2$.
For point $U(-6,6)$:
$x'_U=\frac{1}{3}\times(-6)=-2$, $y'_U=\frac{1}{3}\times6 = 2$.

Step3: Graph the new rectangle

Plot the points $R'(-2,-\frac{2}{3})$, $S'(\frac{8}{3},-\frac{2}{3})$, $T'(\frac{8}{3},2)$, $U'(-2,2)$ and connect them to form the dilated rectangle.

Answer:

Graph the rectangle with vertices $R'(-2,-\frac{2}{3})$, $S'(\frac{8}{3},-\frac{2}{3})$, $T'(\frac{8}{3},2)$, $U'(-2,2)$.