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:

• For point \(E\), the original coordinates are \((0, - 6)\).
• For point \(F\), the original coordinates are \((8,-6)\).
• For point \(G\), the original coordinates are \((8,3)\).
• For point \(H\), the original coordinates are \((0,3)\).

1. Recall the rule for dilation centered at the origin:

• If a point \((x,y)\) is dilated with a scale - factor \(k\) centered at the origin, the new coordinates \((x',y')\) are given by \((x',y')=(k x,k y)\). Here, \(k = \frac{1}{3}\).

1. Calculate the new coordinates for each vertex:

• For point \(E(0,-6)\):
• \(x'=\frac{1}{3}\times0 = 0\).
• \(y'=\frac{1}{3}\times(-6)=-2\). So the new coordinates of \(E\) are \((0,-2)\).
• For point \(F(8,-6)\):
• \(x'=\frac{1}{3}\times8=\frac{8}{3}\).
• \(y'=\frac{1}{3}\times(-6)=-2\). So the new coordinates of \(F\) are \((\frac{8}{3},-2)\).
• For point \(G(8,3)\):
• \(x'=\frac{1}{3}\times8=\frac{8}{3}\).
• \(y'=\frac{1}{3}\times3 = 1\). So the new coordinates of \(G\) are \((\frac{8}{3},1)\).
• For point \(H(0,3)\):
• \(x'=\frac{1}{3}\times0 = 0\).
• \(y'=\frac{1}{3}\times3 = 1\). So the new coordinates of \(H\) are \((0,1)\).

Step1: Identify original coordinates

Identified \(E(0, - 6)\), \(F(8,-6)\), \(G(8,3)\), \(H(0,3)\).

Step2: Recall dilation rule

Used \((x',y')=(k x,k y)\) with \(k=\frac{1}{3}\).

Step3: Calculate new \(E\) coordinates

\(x'=\frac{1}{3}\times0 = 0\), \(y'=\frac{1}{3}\times(-6)=-2\).

Step4: Calculate new \(F\) coordinates

\(x'=\frac{1}{3}\times8=\frac{8}{3}\), \(y'=\frac{1}{3}\times(-6)=-2\).

Step5: Calculate new \(G\) coordinates

\(x'=\frac{1}{3}\times8=\frac{8}{3}\), \(y'=\frac{1}{3}\times3 = 1\).

Step6: Calculate new \(H\) coordinates

\(x'=\frac{1}{3}\times0 = 0\), \(y'=\frac{1}{3}\times3 = 1\).

Answer:

The new coordinates of \(E\) are \((0,-2)\), of \(F\) are \((\frac{8}{3},-2)\), of \(G\) are \((\frac{8}{3},1)\), and of \(H\) are \((0,1)\).