the trapezoid cdef is a dilation of the trapezoid cdef. what is the scale factor of the dilation? simplify…

Explanation:

Step1: Select a corresponding side

Let's consider side $CD$ and $C'D'$. The coordinates of $C(-8,0)$ and $D(8,4)$, the coordinates of $C'(-2,0)$ and $D'(2,1)$.

Step2: Calculate the length of the original - side

Using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, for $CD$ with $x_1=-8,y_1 = 0,x_2 = 8,y_2=4$: [ \begin{align*} CD&=\sqrt{(8 + 8)^2+(4 - 0)^2}\ &=\sqrt{16^2+4^2}\ &=\sqrt{256 + 16}\ &=\sqrt{272}\ &=4\sqrt{17} \end{align*} ]

Step3: Calculate the length of the dilated - side

For $C'D'$ with $x_1=-2,y_1 = 0,x_2 = 2,y_2=1$: [ \begin{align*} C'D'&=\sqrt{(2 + 2)^2+(1 - 0)^2}\ &=\sqrt{4^2+1^2}\ &=\sqrt{16 + 1}\ &=\sqrt{17} \end{align*} ]

Step4: Find the scale factor

The scale factor $k=\frac{C'D'}{CD}=\frac{\sqrt{17}}{4\sqrt{17}}=\frac{1}{4}$

Answer:

$\frac{1}{4}$