Question

write the coordinates of the vertices after a dilation with a scale factor of 1/3, centered at the origin.

Explanation:

Step1: Recall dilation formula

For a dilation centered at the origin with scale - factor $k$, if a point has coordinates $(x,y)$, the new coordinates $(x',y')$ after dilation are given by $(x',y')=(k x,k y)$. Here $k = \frac{1}{3}$.

Step2: Identify original coordinates

Assume the vertices of the rectangle are $E(- 6,0)$, $F(-6,-10)$, $G(4,-10)$, $H(4,0)$.

Step3: Calculate new coordinates for point E

For point $E(-6,0)$:
$x'=\frac{1}{3}\times(-6)=-2$
$y'=\frac{1}{3}\times0 = 0$
New coordinates of $E$ are $(-2,0)$.

Step4: Calculate new coordinates for point F

For point $F(-6,-10)$:
$x'=\frac{1}{3}\times(-6)=-2$
$y'=\frac{1}{3}\times(-10)=-\frac{10}{3}$
New coordinates of $F$ are $(-2,-\frac{10}{3})$.

Step5: Calculate new coordinates for point G

For point $G(4,-10)$:
$x'=\frac{1}{3}\times4=\frac{4}{3}$
$y'=\frac{1}{3}\times(-10)=-\frac{10}{3}$
New coordinates of $G$ are $(\frac{4}{3},-\frac{10}{3})$.

Step6: Calculate new coordinates for point H

For point $H(4,0)$:
$x'=\frac{1}{3}\times4=\frac{4}{3}$
$y'=\frac{1}{3}\times0 = 0$
New coordinates of $H$ are $(\frac{4}{3},0)$.

Answer:

The new coordinates of the vertices are $E(-2,0)$, $F(-2,-\frac{10}{3})$, $G(\frac{4}{3},-\frac{10}{3})$, $H(\frac{4}{3},0)$