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: Analyze side AD length

From the graph, point A is at -4 and D is at 4 on the x - axis. The length of AD is \(|4 - (-4)|=8\) units. So the statement "The length of side AD is 8 units" is true.

Step2: Analyze side A'D' length

Point A' is at -2 and D' is at 2 on the x - axis. The length of A'D' is \(|2-(-2)| = 4\) units. So the statement "The length of side A'D' is 4 units" is true.

Step3: Analyze image - pre - image size

The pre - image (ABCD) has side AD = 8, and the image (A'B'C'D') has side A'D' = 4. Since 4<8, the image is smaller than the pre - image. So the statement "The image is larger than the pre - image" is false.

Step4: Analyze slope of CD and C'D'

Dilation is a similarity transformation, which preserves the slope of lines. Let's find the slope of CD. Let's assume coordinates: C is at (2, y1), D is at (4,0), C' is at (1, y2), D' is at (2,0). The slope of CD: \(m_{CD}=\frac{0 - y1}{4 - 2}=\frac{-y1}{2}\), slope of C'D': \(m_{C'D'}=\frac{0 - y2}{2 - 1}=-y2\). But from the graph, the shape is similar, and the slope should be the same. Wait, let's recalculate. Let's take actual coordinates. Let's say C is (2, 3), D is (4,0), C' is (1, 1.5), D' is (2,0). Slope of CD: \(\frac{0 - 3}{4 - 2}=\frac{-3}{2}=- 1.5\), slope of C'D': \(\frac{0 - 1.5}{2 - 1}=-1.5\). Wait, the option says slope is 2, which is wrong. Wait, maybe I made a mistake. Wait, let's check the x - coordinates. A is at - 4, D at 4; A' at - 2, D' at 2. The scale factor is \(\frac{A'D'}{AD}=\frac{4}{8}=\frac{1}{2}\). Now, for side CD: let's find two points on CD. Let's say C is (2, 3), D is (4,0). The slope is \(\frac{0 - 3}{4 - 2}=\frac{-3}{2}=-1.5\), and for C'D': C' is (1, 1.5), D' is (2,0), slope is \(\frac{0 - 1.5}{2 - 1}=-1.5\), not 2. So the statement "Sides CD and C'D' both have the same slope, 2" is false.

Step5: Analyze scale factor

Scale factor \(k=\frac{\text{length of image side}}{\text{length of pre - image side}}=\frac{A'D'}{AD}=\frac{4}{8}=\frac{1}{2}\). So the statement "The scale factor is \(\frac{1}{2}\)" is true.

Answer:

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