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, the original coordinates of the vertices are \(S(6,6)\), \(T(8,6)\), \(U(8,10)\), \(V(6,10)\).

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 \(x'=k\times x\) and \(y'=k\times y\).
For point \(S(6,6)\):
\(x_S'=\frac{1}{2}\times6 = 3\), \(y_S'=\frac{1}{2}\times6 = 3\), so the new coordinates of \(S\) are \((3,3)\).
For point \(T(8,6)\):
\(x_T'=\frac{1}{2}\times8 = 4\), \(y_T'=\frac{1}{2}\times6 = 3\), so the new coordinates of \(T\) are \((4,3)\).
For point \(U(8,10)\):
\(x_U'=\frac{1}{2}\times8 = 4\), \(y_U'=\frac{1}{2}\times10 = 5\), so the new coordinates of \(U\) are \((4,5)\).
For point \(V(6,10)\):
\(x_V'=\frac{1}{2}\times6 = 3\), \(y_V'=\frac{1}{2}\times10 = 5\), so the new coordinates of \(V\) are \((3,5)\).

Answer:

The new coordinates of the vertices are \(S'(3,3)\), \(T'(4,3)\), \(U'(4,5)\), \(V'(3,5)\)