Question

the triangle def is a dilation of the triangle def. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole

Explanation:

Step1: Find the length of a side in the original triangle

The length of \(EF\) in \(\triangle DEF\). The \(x -\)coordinates of \(E(2,9)\) and \(F(- 5,9)\). Using the distance formula for horizontal line \(d=\vert x_2 - x_1\vert\), \(EF=\vert2-(-5)\vert = 7\).

Step2: Find the length of the corresponding side in the dilated triangle

The length of \(E'F'\) in \(\triangle D'E'F'\). The \(x -\)coordinates of \(E'(1,3)\) and \(F'(-2,3)\). Using the distance formula for horizontal line \(d = \vert x_2 - x_1\vert\), \(E'F'=\vert1-(-2)\vert=3\). Wait, no, better to use the ratio of vertical or horizontal segments. Let's use the \(y -\)coordinate difference for a vertical segment.
Take point \(D(-3,-9)\) and \(D'(-1,-3)\). The \(y -\)coordinate of \(D\) is \(y_D=-9\), the \(y -\)coordinate of \(D'\) is \(y_{D'}=-3\). The scale factor \(k\) of dilation is given by the formula \(k=\frac{\text{length of segment in image}}{\text{length of segment in pre - image}}\).
\(k=\frac{\vert y_{D'}-y_{E'}\vert}{\vert y_D - y_E\vert}\). \(y_{E'}=3\), \(y_E = 9\), \(y_{D'}=-3\), \(y_D=-9\). \(\vert y_{D'}-y_{E'}\vert=\vert-3 - 3\vert = 6\), \(\vert y_D - y_E\vert=\vert-9 - 9\vert=18\).

Step3: Calculate the scale factor

Scale factor \(k=\frac{6}{18}=\frac{1}{3}\)

Answer:

\(\frac{1}{3}\)