Question

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

Explanation:

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

• Assume the vertices of the rectangle are \(B(- 10,0)\), \(C(-10,5)\), \(D(10,5)\), and \(E(10,0)\).
• The formula for dilation centered at the origin with a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{5}\).

1. Calculate the new coordinates of vertex \(B\):

• For \(B(-10,0)\), when applying the dilation formula \(x=-10\) and \(y = 0\), \(x'=\frac{1}{5}\times(-10)=-2\) and \(y'=\frac{1}{5}\times0 = 0\). So the new coordinates of \(B\) are \(B'(-2,0)\).

1. Calculate the new coordinates of vertex \(C\):

• For \(C(-10,5)\), \(x=-10\) and \(y = 5\). Then \(x'=\frac{1}{5}\times(-10)=-2\) and \(y'=\frac{1}{5}\times5 = 1\). So the new coordinates of \(C\) are \(C'(-2,1)\).

1. Calculate the new coordinates of vertex \(D\):

• For \(D(10,5)\), \(x = 10\) and \(y = 5\). Then \(x'=\frac{1}{5}\times10 = 2\) and \(y'=\frac{1}{5}\times5=1\). So the new coordinates of \(D\) are \(D'(2,1)\).

1. Calculate the new coordinates of vertex \(E\):

• For \(E(10,0)\), \(x = 10\) and \(y = 0\). Then \(x'=\frac{1}{5}\times10 = 2\) and \(y'=\frac{1}{5}\times0 = 0\). So the new coordinates of \(E\) are \(E'(2,0)\).

Answer:

The new coordinates of the vertices are \(B'(-2,0)\), \(C'(-2,1)\), \(D'(2,1)\), \(E'(2,0)\)