Question

the parallelogram ( defg ) is a dilation of the parallelogram ( defg ). what is the scale factor of the dilation?
simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find the length of corresponding sides

Let's consider the horizontal distance from \(E\) to \(D\) and \(E'\) to \(D'\).
The \(x\) - coordinate of \(E=(3,1)\) and \(D=(3,2)\), the length \(ED = 2 - 1=1\).
The \(x\) - coordinate of \(E'=(9,0)\) and \(D'=(9,6)\), the length \(E'D'=6 - 0 = 6\).
Another way: consider the distance from \(G\) to \(F\) and \(G'\) to \(F'\). \(G=(- 3,-2)\), \(F=(-3,-3)\), so \(GF=|-2-(-3)| = 1\). \(G'=(-9,-8)\), \(F'=(-9,-9)\), so \(G'F'=|-8-(-9)| = 1\). But if we use the distance from \(E\) to \(D\) (vertical side) and \(E'\) to \(D'\) (vertical side)
The formula for the scale factor \(k\) of a dilation is \(k=\frac{\text{length of side in image}}{\text{length of corresponding side in pre - image}}\)
Take the vertical side: \(ED\) (pre - image) has length \(2 - 1=1\) and \(E'D'\) (image) has length \(6-0 = 6\)

Step2: Calculate the scale factor

Using the formula \(k=\frac{\text{length of side in image}}{\text{length of corresponding side in pre - image}}\), if we take the vertical sides. Let the pre - image be \(DEFG\) and the image be \(D'E'F'G'\).
\(k=\frac{E'D'}{ED}\)
Since \(E'D' = 6\) and \(ED=1\), \(k = 3\) (if we consider the horizontal distance from \(E=(3,1)\) to \(D=(3,2)\) (length \(1\)) and \(E'=(9,0)\) to \(D'=(9,6)\) (length \(6\)), or we can also use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For a vertical line \(x_1=x_2\), \(d = |y_2 - y_1|\).
Another approach: use the ratio of \(x\) - coordinates (since the center of dilation is likely the origin, assume the dilation is centered at the origin). For point \(E(3,1)\) and \(E'(9,0)\) (using the non - zero coordinate for non - vertical/horizontal lines in a more general sense). If we assume the dilation rule \((x,y)\to(kx,ky)\). Taking \(x\) - coordinates: \(3k=9\) (for the \(x\) - coordinate of \(E\) and \(E'\)), solving for \(k\) gives \(k = 3\)

Answer:

\(3\)