Question

write the coordinates of the vertices after a dilation with a scale factor of 1/3, centered at the origin.

Explanation:

1. First, assume the coordinates of the vertices of the original polygon:

• Let's assume the coordinates of point \(C\) are \((0, - 6)\), point \(D\) are \((-4,-3)\), point \(E\) are \((-8, - 6)\) (we need to visually estimate from the graph).
• The rule for dilation centered at the origin with a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{3}\).

1. Calculate the new coordinates for each vertex:

• For point \(C(0,-6)\):
• Using the dilation formula \((x,y)\to(kx,ky)\) with \(k=\frac{1}{3}\), \(x = 0\) and \(y=-6\).
• The new \(x\) - coordinate is \(x_{new}=k\times x=\frac{1}{3}\times0 = 0\).
• The new \(y\) - coordinate is \(y_{new}=k\times y=\frac{1}{3}\times(-6)=-2\). So the new coordinates of \(C\) are \((0, - 2)\).
• For point \(D(-4,-3)\):
• The new \(x\) - coordinate is \(x_{new}=k\times x=\frac{1}{3}\times(-4)=-\frac{4}{3}\).
• The new \(y\) - coordinate is \(y_{new}=k\times y=\frac{1}{3}\times(-3)=-1\). So the new coordinates of \(D\) are \((-\frac{4}{3},-1)\).
• For point \(E(-8,-6)\):
• The new \(x\) - coordinate is \(x_{new}=k\times x=\frac{1}{3}\times(-8)=-\frac{8}{3}\).
• The new \(y\) - coordinate is \(y_{new}=k\times y=\frac{1}{3}\times(-6)=-2\). So the new coordinates of \(E\) are \((-\frac{8}{3},-2)\).

Answer:

The new coordinates of \(C\) are \((0, - 2)\), the new coordinates of \(D\) are \((-\frac{4}{3},-1)\), and the new coordinates of \(E\) are \((-\frac{8}{3},-2)\) (assuming the original coordinates of \(C\), \(D\), and \(E\) as estimated above).