Question

identifying vertices of the pre - image of a dilation
triangle xyz was dilated using the rule ( d_{o,0.25}(x,y)\to(0.25x,0.25y) ).
the image is shown in the diagram. what are the coordinates of z of the pre - image?
( (-8,-4) )
( (-8,2) )
( (-2,-1) )
( (-0.25,-0.5) )

Explanation:

Step1: Determine the coordinates of \(Z'\)

From the graph, the coordinates of \(Z'\) (image) are \((- 2,-1)\).

Step2: Use the inverse of the dilation rule

The dilation rule is \(D_{O,0.25}(x,y)=(0.25x,0.25y)\). To find the pre - image \((x,y)\) from the image \((x',y')\), we use the inverse rule \(x=\frac{x'}{0.25}\), \(y = \frac{y'}{0.25}\) (since \(x'=0.25x\) and \(y'=0.25y\)).
Substitute \(x'=-2\) and \(y'=-1\) into the inverse formulas:
For the \(x\) - coordinate of the pre - image: \(x=\frac{-2}{0.25}=\frac{-2}{\frac{1}{4}}=-2\times4 = - 8\)
For the \(y\) - coordinate of the pre - image: \(y=\frac{-1}{0.25}=\frac{-1}{\frac{1}{4}}=-1\times4=-4\)

Answer:

\((-8,-4)\)