Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{3}$, centered at the origin.

Explanation:

Step1: Identify original coordinates

Let's assume the original coordinates of the vertices are \(D(-6,-9)\), \(E(3,-9)\), \(F(3, - 3)\), \(G(-6,-3)\)

Step2: Apply dilation formula

For a dilation centered at the origin with scale - factor \(k=\frac{1}{3}\), the formula to find the new coordinates \((x',y')\) of a point \((x,y)\) is \(x' = kx\) and \(y'=ky\).
For point \(D(-6,-9)\):
\(x_D'=\frac{1}{3}\times(-6)= - 2\)
\(y_D'=\frac{1}{3}\times(-9)=-3\)
So \(D'(-2,-3)\)
For point \(E(3,-9)\):
\(x_E'=\frac{1}{3}\times3 = 1\)
\(y_E'=\frac{1}{3}\times(-9)=-3\)
So \(E'(1,-3)\)
For point \(F(3,-3)\):
\(x_F'=\frac{1}{3}\times3 = 1\)
\(y_F'=\frac{1}{3}\times(-3)=-1\)
So \(F'(1,-1)\)
For point \(G(-6,-3)\):
\(x_G'=\frac{1}{3}\times(-6)=-2\)
\(y_G'=\frac{1}{3}\times(-3)=-1\)
So \(G'(-2,-1)\)

Answer:

\(D'(-2,-3)\)
\(E'(1,-3)\)
\(F'(1,-1)\)
\(G'(-2,-1)\)