Question

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

Explanation:

Step1: Find coordinates of vertices

First, identify the coordinates of the rectangle's vertices. From the graph:

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

Step2: Apply dilation formula

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

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

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

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

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

Step3: Graph the new vertices

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

Answer:

The dilated rectangle has vertices at \( S'(-2, 1) \), \( T'(2, 1) \), \( U'(2, 2) \), \( V'(-2, 2) \) (graph by plotting these points and connecting them).