Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{2}$, centered at the origin.

Explanation:

Step1: Identify original coordinates

From the graph, we can see that the coordinates of the vertices are \(A(-6,-8)\), \(B(-6, - 2)\), \(C(-2,8)\), \(D(-2,-8)\).

Step2: Apply dilation formula

For a dilation centered at the origin with scale - factor \(k=\frac{1}{2}\), the formula to find the new coordinates \((x',y')\) of a point \((x,y)\) is \(x' = kx\) and \(y'=ky\).
For point \(A(-6,-8)\):
\(x'_A=\frac{1}{2}\times(-6)=-3\)
\(y'_A=\frac{1}{2}\times(-8)=-4\)
For point \(B(-6,-2)\):
\(x'_B=\frac{1}{2}\times(-6)=-3\)
\(y'_B=\frac{1}{2}\times(-2)=-1\)
For point \(C(-2,8)\):
\(x'_C=\frac{1}{2}\times(-2)=-1\)
\(y'_C=\frac{1}{2}\times8 = 4\)
For point \(D(-2,-8)\):
\(x'_D=\frac{1}{2}\times(-2)=-1\)
\(y'_D=\frac{1}{2}\times(-8)=-4\)

Answer:

\(A'(-3,-4)\)
\(B'(-3,-1)\)
\(C'(-1,4)\)
\(D'(-1,-4)\)