Question

the trapezoid ( ijkl ) is a dilation of the trapezoid ( ijkl ). 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: Identify corresponding side lengths

Find the length of a side in the original trapezoid \(IJKL\) and the corresponding side in the dilated trapezoid \(I'J'K'L'\). Let's use the side \(IJ\) (original) and \(I'J'\) (dilated).
The length of \(IJ\): The \(x -\)coordinates of \(I(-8,0)\) and \(J(8,0)\). Using the distance formula \(d=\vert x_2 - x_1\vert\) (since \(y - \)coordinates are \(0\)), \(IJ=\vert8-(-8)\vert=\vert8 + 8\vert = 16\).
The length of \(I'J'\): The \(x -\)coordinates of \(I'(-2,0)\) and \(J'(2,0)\). Using the distance formula \(d=\vert x_2 - x_1\vert\) (since \(y - \)coordinates are \(0\)), \(I'J'=\vert2-(-2)\vert=\vert2 + 2\vert=4\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\).
\(k=\frac{I'J'}{IJ}\)
Substitute \(I'J' = 4\) and \(IJ = 16\) into the formula: \(k=\frac{4}{16}\)
Simplify the fraction \(\frac{4}{16}=\frac{4\div4}{16\div4}=\frac{1}{4}\)

Answer:

\(\frac{1}{4}\)