Question

which statements are true about the trapezoids? choose three correct answers.
the length of side ad is 8 units.
the image is larger than the pre - image.
the length of side ad is 4 units.
the scale factor is \frac{1}{2}.
sides cd and cd both have the same slope. 2

Explanation:

Step1: Calculate the length of \(AD\)

From the graph, \(A=(0, - 4)\) and \(D=(0,4)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(x_1 = x_2=0\), \(y_1=-4\), \(y_2 = 4\), we have \(AD=\vert-4 - 4\vert=8\) units.

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

From the graph, \(A'=(0,-2)\) and \(D'=(0,2)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(x_1 = x_2=0\), \(y_1=-2\), \(y_2 = 2\), we have \(A'D'=\vert-2 - 2\vert=4\) units.

Step3: Calculate the scale factor

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

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

Let \(C=(4,2)\), \(D=(0,4)\), the slope of \(CD\) is \(m_{CD}=\frac{4 - 2}{0 - 4}=-\frac{1}{2}\). Let \(C'=(2,1)\), \(D'=(0,2)\), the slope of \(C'D'\) is \(m_{C'D'}=\frac{2 - 1}{0 - 2}=-\frac{1}{2}\).

Answer:

The length of side \(AD\) is \(8\) units, the length of side \(A'D'\) is \(4\) units, the scale factor is \(\frac{1}{2}\), sides \(CD\) and \(C'D'\) both have the same slope. So the correct statements are: "The length of side \(AD\) is \(8\) units", "The length of side \(A'D'\) is \(4\) units", "The scale factor is \(\frac{1}{2}\)", "Sides \(CD\) and \(C'D'\) both have the same slope".