Question

cai uses a computer program to resize photos for the online school newspaper. rectangle defg represents the photo. she drags the top right corner of the rectangle along the dashed line to form the image rectangle defg. the center of dilation is at the origin. the scale factor is $\frac{1}{2}$. a. what are the coordinates of the vertices of image rectangle defg? show your work.

Explanation:

Step1: Identify original coordinates

Assume the coordinates of rectangle $DEFG$ are $D(0,0)$, $E(0,8)$, $F(10,8)$, $G(10,0)$.

Step2: Apply dilation formula

For a point $(x,y)$ dilated with a scale - factor $k$ centered at the origin, the new coordinates $(x',y')$ are given by $(x',y')=(k x,k y)$. Here $k = \frac{1}{2}$.
For point $D(0,0)$:
$x'=\frac{1}{2}\times0 = 0$, $y'=\frac{1}{2}\times0 = 0$, so $D'(0,0)$.
For point $E(0,8)$:
$x'=\frac{1}{2}\times0 = 0$, $y'=\frac{1}{2}\times8 = 4$, so $E'(0,4)$.
For point $F(10,8)$:
$x'=\frac{1}{2}\times10 = 5$, $y'=\frac{1}{2}\times8 = 4$, so $F'(5,4)$.
For point $G(10,0)$:
$x'=\frac{1}{2}\times10 = 5$, $y'=\frac{1}{2}\times0 = 0$, so $G'(5,0)$.

Answer:

$D'(0,0)$, $E'(0,4)$, $F'(5,4)$, $G'(5,0)$