Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{4}$, centered at the origin.

Explanation:

Step1: Identify original coordinates

Let's assume the coordinates of the vertices are \(S(-8,0)\), \(T(-4,8)\), \(U(-4,0)\), \(R(-8, - 8)\) 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 \(S(-8,0)\):
\(x_S'=\frac{1}{4}\times(-8)= - 2\)
\(y_S'=\frac{1}{4}\times0 = 0\)
So the new coordinates of \(S\) are \(S'(-2,0)\)
For point \(T(-4,8)\):
\(x_T'=\frac{1}{4}\times(-4)=-1\)
\(y_T'=\frac{1}{4}\times8 = 2\)
So the new coordinates of \(T\) are \(T'(-1,2)\)
For point \(U(-4,0)\):
\(x_U'=\frac{1}{4}\times(-4)=-1\)
\(y_U'=\frac{1}{4}\times0 = 0\)
So the new coordinates of \(U\) are \(U'(-1,0)\)
For point \(R(-8,-8)\):
\(x_R'=\frac{1}{4}\times(-8)=-2\)
\(y_R'=\frac{1}{4}\times(-8)=-2\)
So the new coordinates of \(R\) are \(R'(-2,-2)\)

Answer:

The new coordinates of the vertices are \(S'(-2,0)\), \(T'(-1,2)\), \(U'(-1,0)\), \(R'(-2,-2)\)