Question

graph the image of rectangle stuv after a dilation with a scale factor of \\(\frac{1}{4}\\), centered at the origin.

Explanation:

Step1: Identify coordinates of vertices

First, find the coordinates of the vertices of rectangle \( STUV \):

• \( S(-8, 4) \)
• \( T(8, 4) \)
• \( U(8, 8) \)
• \( V(-8, 8) \)

Step2: Apply dilation formula

The dilation formula centered at the origin with scale factor \( k \) is \( (x, y) \to (kx, ky) \). Here, \( k = \frac{1}{4} \).

For \( S(-8, 4) \):
\( x' = -8 \times \frac{1}{4} = -2 \)
\( y' = 4 \times \frac{1}{4} = 1 \)
So, \( S'(-2, 1) \)

For \( T(8, 4) \):
\( x' = 8 \times \frac{1}{4} = 2 \)
\( y' = 4 \times \frac{1}{4} = 1 \)
So, \( T'(2, 1) \)

For \( U(8, 8) \):
\( x' = 8 \times \frac{1}{4} = 2 \)
\( y' = 8 \times \frac{1}{4} = 2 \)
So, \( U'(2, 2) \)

For \( V(-8, 8) \):
\( x' = -8 \times \frac{1}{4} = -2 \)
\( y' = 8 \times \frac{1}{4} = 2 \)
So, \( V'(-2, 2) \)

Step3: Graph the new vertices

Plot the points \( S'(-2, 1) \), \( T'(2, 1) \), \( U'(2, 2) \), and \( V'(-2, 2) \) on the coordinate plane and connect them to form the dilated rectangle.

Answer:

The dilated rectangle has vertices at \( S'(-2, 1) \), \( T'(2, 1) \), \( U'(2, 2) \), and \( V'(-2, 2) \). (To graph, plot these points and connect them in order.)