Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{5}$, centered at the origin.

Explanation:

Step1: Identify original coordinates

The original coordinates of the vertices are \(P(0, - 10)\), \(Q(0,5)\), \(R(5,5)\), \(S(5,-10)\).

Step2: Apply dilation formula

For a dilation with scale - factor \(k=\frac{1}{5}\) centered at the origin \((x,y)\to(kx,ky)\).
For point \(P(0,-10)\):
\(x = 0\), \(y=-10\), new \(x_1=\frac{1}{5}\times0 = 0\), new \(y_1=\frac{1}{5}\times(-10)=-2\), so the new coordinate of \(P\) is \((0,-2)\).
For point \(Q(0,5)\):
\(x = 0\), \(y = 5\), new \(x_2=\frac{1}{5}\times0=0\), new \(y_2=\frac{1}{5}\times5 = 1\), so the new coordinate of \(Q\) is \((0,1)\).
For point \(R(5,5)\):
\(x = 5\), \(y = 5\), new \(x_3=\frac{1}{5}\times5 = 1\), new \(y_3=\frac{1}{5}\times5=1\), so the new coordinate of \(R\) is \((1,1)\).
For point \(S(5,-10)\):
\(x = 5\), \(y=-10\), new \(x_4=\frac{1}{5}\times5 = 1\), new \(y_4=\frac{1}{5}\times(-10)=-2\), so the new coordinate of \(S\) is \((1,-2)\).

Answer:

The new coordinates of the vertices are \(P(0,-2)\), \(Q(0,1)\), \(R(1,1)\), \(S(1,-2)\)