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 rectangle vertices

Let's assume the coordinates of rectangle \(STUV\) are \(S(- 8,4)\), \(T(8,4)\), \(U(8,8)\), \(V(-8,8)\) from the graph.

Step2: Apply dilation formula

For a dilation centered at the origin with scale - factor \(k=\frac{1}{4}\), the formula to find the new coordinates \((x',y')\) of a point \((x,y)\) is \(x' = kx\) and \(y'=ky\).
For point \(S(-8,4)\):
\(x_S'=\frac{1}{4}\times(-8)= - 2\) and \(y_S'=\frac{1}{4}\times4 = 1\)
For point \(T(8,4)\):
\(x_T'=\frac{1}{4}\times8 = 2\) and \(y_T'=\frac{1}{4}\times4=1\)
For point \(U(8,8)\):
\(x_U'=\frac{1}{4}\times8 = 2\) and \(y_U'=\frac{1}{4}\times8 = 2\)
For point \(V(-8,8)\):
\(x_V'=\frac{1}{4}\times(-8)=-2\) and \(y_V'=\frac{1}{4}\times8 = 2\)

Step3: Graph new rectangle

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 new rectangle has vertices \(S'(-2,1)\), \(T'(2,1)\), \(U'(2,2)\), \(V'(-2,2)\) which should be graphed on the coordinate - plane.