Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{3}$, centered at the origin.

Explanation:

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

• Assume the coordinates of point \(C\) are \((0, - 6)\), point \(D\) are \((-4,-2)\), point \(E\) are \((-8,-6)\), and point \(B\) are \((-6,-8)\).
• 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 \(C(0, - 6)\):
• Multiply the \(x\) - coordinate and \(y\) - coordinate by the scale factor \(\frac{1}{3}\).
• \(x_{new}=0\times\frac{1}{3}=0\), \(y_{new}=-6\times\frac{1}{3}=-2\). So the new coordinates of \(C\) are \((0, - 2)\).
• For point \(D(-4,-2)\):
• \(x_{new}=-4\times\frac{1}{3}=-\frac{4}{3}\), \(y_{new}=-2\times\frac{1}{3}=-\frac{2}{3}\). So the new coordinates of \(D\) are \((-\frac{4}{3},-\frac{2}{3})\).
• For point \(E(-8,-6)\):
• \(x_{new}=-8\times\frac{1}{3}=-\frac{8}{3}\), \(y_{new}=-6\times\frac{1}{3}=-2\). So the new coordinates of \(E\) are \((-\frac{8}{3},-2)\).
• For point \(B(-6,-8)\):
• \(x_{new}=-6\times\frac{1}{3}=-2\), \(y_{new}=-8\times\frac{1}{3}=-\frac{8}{3}\). So the new coordinates of \(B\) are \((-2,-\frac{8}{3})\).

Answer:

The new coordinates of \(C\) are \((0, - 2)\), of \(D\) are \((-\frac{4}{3},-\frac{2}{3})\), of \(E\) are \((-\frac{8}{3},-2)\), and of \(B\) are \((-2,-\frac{8}{3})\).