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 original coordinates of the vertices are \(D(-4,0)\), \(E(-4,8)\), \(F(6, - 4)\), \(G(6,-8)\).

Step2: Apply dilation formula

For a dilation centered at the origin with scale - factor \(k=\frac{1}{4}\), the formula for the new coordinates \((x',y')\) of a point \((x,y)\) is \((x',y')=(k x,k y)\).
For point \(D(-4,0)\):
\(x'=\frac{1}{4}\times(-4)= - 1\), \(y'=\frac{1}{4}\times0 = 0\), so the new coordinates of \(D\) are \((-1,0)\).
For point \(E(-4,8)\):
\(x'=\frac{1}{4}\times(-4)=-1\), \(y'=\frac{1}{4}\times8 = 2\), so the new coordinates of \(E\) are \((-1,2)\).
For point \(F(6,-4)\):
\(x'=\frac{1}{4}\times6=\frac{3}{2}\), \(y'=\frac{1}{4}\times(-4)=-1\), so the new coordinates of \(F\) are \((\frac{3}{2},-1)\).
For point \(G(6,-8)\):
\(x'=\frac{1}{4}\times6=\frac{3}{2}\), \(y'=\frac{1}{4}\times(-8)=-2\), so the new coordinates of \(G\) are \((\frac{3}{2},-2)\).

Answer:

The new coordinates of \(D\) are \((-1,0)\), of \(E\) are \((-1,2)\), of \(F\) are \((\frac{3}{2},-1)\), and of \(G\) are \((\frac{3}{2},-2)\)