Question

graph the image of rectangle rstu after a dilation with a scale factor of $\frac{1}{5}$, centered at the origin.

Explanation:

Step1: Find the coordinates of the original rectangle

Assume the coordinates of rectangle \(RSTU\) are \(U(- 10,5)\), \(T(10,5)\), \(S(10,-10)\), \(R(-10,-10)\) (by observing the graph).

Step2: Apply the dilation formula

The formula for dilation centered at the origin is \((x,y)\to(kx,ky)\), where \(k = \frac{1}{5}\).
For point \(U(-10,5)\):
\(x'=\frac{1}{5}\times(-10)=- 2\), \(y'=\frac{1}{5}\times5 = 1\), so \(U'(-2,1)\)
For point \(T(10,5)\):
\(x'=\frac{1}{5}\times10 = 2\), \(y'=\frac{1}{5}\times5=1\), so \(T'(2,1)\)
For point \(S(10,-10)\):
\(x'=\frac{1}{5}\times10 = 2\), \(y'=\frac{1}{5}\times(-10)=-2\), so \(S'(2,-2)\)
For point \(R(-10,-10)\):
\(x'=\frac{1}{5}\times(-10)=-2\), \(y'=\frac{1}{5}\times(-10)=-2\), so \(R'(-2,-2)\)

Step3: Plot the new points

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

Answer:

The coordinates of the vertices of the dilated rectangle are \(U'(-2,1)\), \(T'(2,1)\), \(S'(2,-2)\), \(R'(-2,-2)\). Graph these points and connect them to get the image of rectangle \(RSTU\) after dilation.