Question

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

Explanation:

1. First, identify the original coordinates of the vertices:

• From the graph, the coordinates of point \(T\) are \((- 8,2)\), the coordinates of point \(U\) are \((4,2)\), the coordinates of point \(V\) are \((4,-10)\), and the coordinates of point \(S\) are \((-8,-10)\).

1. Then, use the dilation formula for a dilation centered at the origin with a scale - factor \(k\). The formula is \((x,y)\to(kx,ky)\) where \((x,y)\) are the original coordinates and \((kx,ky)\) are the coordinates after dilation, and \(k = \frac{1}{2}\) in this case.

• For point \(T(-8,2)\):
• Calculate the new \(x\) - coordinate: \(x'=\frac{1}{2}\times(-8)=-4\).
• Calculate the new \(y\) - coordinate: \(y'=\frac{1}{2}\times2 = 1\). So the new coordinates of \(T\) are \((-4,1)\).
• For point \(U(4,2)\):
• Calculate the new \(x\) - coordinate: \(x'=\frac{1}{2}\times4 = 2\).
• Calculate the new \(y\) - coordinate: \(y'=\frac{1}{2}\times2 = 1\). So the new coordinates of \(U\) are \((2,1)\).
• For point \(V(4,-10)\):
• Calculate the new \(x\) - coordinate: \(x'=\frac{1}{2}\times4 = 2\).
• Calculate the new \(y\) - coordinate: \(y'=\frac{1}{2}\times(-10)=-5\). So the new coordinates of \(V\) are \((2,-5)\).
• For point \(S(-8,-10)\):
• Calculate the new \(x\) - coordinate: \(x'=\frac{1}{2}\times(-8)=-4\).
• Calculate the new \(y\) - coordinate: \(y'=\frac{1}{2}\times(-10)=-5\). So the new coordinates of \(S\) are \((-4,-5)\).

Answer:

The coordinates of \(T\) after dilation are \((-4,1)\), the coordinates of \(U\) after dilation are \((2,1)\), the coordinates of \(V\) after dilation are \((2,-5)\), and the coordinates of \(S\) after dilation are \((-4,-5)\).