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

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

Answer

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

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) )

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

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).