Question

trapezoid abcd was dilated to create trapezoid abcd.
which statements are true about the trapezoids? select three options.
the length of side ad is 8 units.
the length of side ad is 4 units.
the image is larger than the pre - image.
sides cd and cd both have the same slope, 2.
the scale factor is \\( \frac { 1 } { 2 } \\).

Explanation:

Step1: Calculate the length of AD

Using the distance formula for points \(A(-4,0)\) and \(D(4,0)\). Since \(y - coordinates\) are the same (\(y = 0\)), the distance \(AD=\vert x_{D}-x_{A}\vert=\vert4 - (-4)\vert = 8\) units.

Step2: Calculate the length of \(A'D'\)

For points \(A'(-2,0)\) and \(D'(2,0)\), \(A'D'=\vert x_{D'}-x_{A'}\vert=\vert2-(-2)\vert = 4\) units.

Step3: Determine the scale factor

The scale factor \(k=\frac{A'D'}{AD}=\frac{4}{8}=\frac{1}{2}\)

Step4: Calculate the slope of \(CD\) and \(C'D'\)

For \(C(2,1)\) and \(D(4,0)\), slope \(m_{CD}=\frac{0 - 1}{4 - 2}=-\frac{1}{2}\). For \(C'(1,0.5)\) and \(D'(2,0)\), slope \(m_{C'D'}=\frac{0 - 0.5}{2 - 1}=-\frac{1}{2}\). Also, since the scale factor \(k = \frac{1}{2}<1\), the image (\(A'B'C'D'\)) is smaller than the pre - image (\(ABCD\))

Answer:

The length of side \(AD\) is \(8\) units, The length of side \(A'D'\) is \(4\) units, The scale factor is \(\frac{1}{2}\)