Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{4}$, centered at the origin.

Explanation:

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

• From the graph, assume the coordinates of the vertices of the rectangle are \(T(- 8,-8)\), \(U(8,-8)\), \(V(8,-4)\), \(W(-8,-4)\).
• The rule for a dilation centered at the origin with a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{4}\).

1. Calculate the new coordinates for each vertex:

• For vertex \(T(-8,-8)\):
• \(x\) - coordinate of the new vertex \(T'\): \(x'=\frac{1}{4}\times(-8)=-2\).
• \(y\) - coordinate of the new vertex \(T'\): \(y'=\frac{1}{4}\times(-8)=-2\). So, \(T'(-2,-2)\).
• For vertex \(U(8,-8)\):
• \(x\) - coordinate of the new vertex \(U'\): \(x'=\frac{1}{4}\times8 = 2\).
• \(y\) - coordinate of the new vertex \(U'\): \(y'=\frac{1}{4}\times(-8)=-2\). So, \(U'(2,-2)\).
• For vertex \(V(8,-4)\):
• \(x\) - coordinate of the new vertex \(V'\): \(x'=\frac{1}{4}\times8 = 2\).
• \(y\) - coordinate of the new vertex \(V'\): \(y'=\frac{1}{4}\times(-4)=-1\). So, \(V'(2,-1)\).
• For vertex \(W(-8,-4)\):
• \(x\) - coordinate of the new vertex \(W'\): \(x'=\frac{1}{4}\times(-8)=-2\).
• \(y\) - coordinate of the new vertex \(W'\): \(y'=\frac{1}{4}\times(-4)=-1\). So, \(W'(-2,-1)\).

Answer:

The coordinates of the vertices after dilation are \(T'(-2,-2)\), \(U'(2,-2)\), \(V'(2,-1)\), \(W'(-2,-1)\)