Question

write the coordinates of the vertices after a dilation with a scale factor of ¼, centered at the origin.

Explanation:

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

• From the graph, the coordinates of point \(T\) are \((- 8,-8)\), the coordinates of point \(W\) are \((-8, - 4)\), the coordinates of point \(V\) are \((8,-4)\), and the coordinates of point \(U\) are \((8,-8)\).

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, \(k = \frac{1}{4}\).

1. Calculate the new coordinates for each vertex:

• For point \(T(-8,-8)\):
• \(x'=\frac{1}{4}\times(-8)=-2\)
• \(y'=\frac{1}{4}\times(-8)=-2\)
• The new coordinates of \(T\) are \((-2,-2)\).
• For point \(W(-8, - 4)\):
• \(x'=\frac{1}{4}\times(-8)=-2\)
• \(y'=\frac{1}{4}\times(-4)=-1\)
• The new coordinates of \(W\) are \((-2,-1)\).
• For point \(V(8,-4)\):
• \(x'=\frac{1}{4}\times8 = 2\)
• \(y'=\frac{1}{4}\times(-4)=-1\)
• The new coordinates of \(V\) are \((2,-1)\).
• For point \(U(8,-8)\):
• \(x'=\frac{1}{4}\times8 = 2\)
• \(y'=\frac{1}{4}\times(-8)=-2\)
• The new coordinates of \(U\) are \((2,-2)\).

Step1: Identify original coordinates

\(T(-8,-8),W(-8,-4),V(8,-4),U(8,-8)\)

Step2: Apply dilation formula

For a point \((x,y)\) with \(k = \frac{1}{4}\), new point is \((\frac{1}{4}x,\frac{1}{4}y)\)

Step3: Calculate new coordinates for \(T\)

\(x_T'=\frac{1}{4}\times(-8)=-2,y_T'=\frac{1}{4}\times(-8)=-2\)

Step4: Calculate new coordinates for \(W\)

\(x_W'=\frac{1}{4}\times(-8)=-2,y_W'=\frac{1}{4}\times(-4)=-1\)

Step5: Calculate new coordinates for \(V\)

\(x_V'=\frac{1}{4}\times8 = 2,y_V'=\frac{1}{4}\times(-4)=-1\)

Step6: Calculate new coordinates for \(U\)

\(x_U'=\frac{1}{4}\times8 = 2,y_U'=\frac{1}{4}\times(-8)=-2\)

Answer:

The new coordinates of \(T\) are \((-2,-2)\), the new coordinates of \(W\) are \((-2,-1)\), the new coordinates of \(V\) are \((2,-1)\), and the new coordinates of \(U\) are \((2,-2)\).