Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{2}$, centered at the origin.

Explanation:

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

• From the graph, assume the coordinates of point \(E\) are \((- 6,-8)\), point \(F\) are \((2,-8)\), point \(G\) are \((2,-6)\), and point \(H\) are \((-6,-6)\).
• The rule for dilation centered at the origin with a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{2}\).

1. Calculate the new coordinates for each vertex:

• For point \(E(-6,-8)\):
• \(x\) - coordinate of the new point \(E'\): \(x'=\frac{1}{2}\times(-6)=-3\).
• \(y\) - coordinate of the new point \(E'\): \(y'=\frac{1}{2}\times(-8)=-4\). So, \(E'(-3,-4)\).
• For point \(F(2,-8)\):
• \(x\) - coordinate of the new point \(F'\): \(x'=\frac{1}{2}\times2 = 1\).
• \(y\) - coordinate of the new point \(F'\): \(y'=\frac{1}{2}\times(-8)=-4\). So, \(F'(1,-4)\).
• For point \(G(2,-6)\):
• \(x\) - coordinate of the new point \(G'\): \(x'=\frac{1}{2}\times2 = 1\).
• \(y\) - coordinate of the new point \(G'\): \(y'=\frac{1}{2}\times(-6)=-3\). So, \(G'(1,-3)\).
• For point \(H(-6,-6)\):
• \(x\) - coordinate of the new point \(H'\): \(x'=\frac{1}{2}\times(-6)=-3\).
• \(y\) - coordinate of the new point \(H'\): \(y'=\frac{1}{2}\times(-6)=-3\). So, \(H'(-3,-3)\).

Step1: Identify original coordinates

Assume \(E(-6,-8)\), \(F(2,-8)\), \(G(2,-6)\), \(H(-6,-6)\)

Step2: Apply dilation rule

Use \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\) for each point

Step3: Calculate new \(x\) - coordinates

Multiply original \(x\) - values by \(\frac{1}{2}\)

Step4: Calculate new \(y\) - coordinates

Multiply original \(y\) - values by \(\frac{1}{2}\)

Answer:

The coordinates of the vertices after dilation are \(E'(-3,-4)\), \(F'(1,-4)\), \(G'(1,-3)\), \(H'(-3,-3)\)