Question

ashti is designing a rug that uses shapes to tell a story, similar to what rug weavers in her culture have done for generations. she uses a computer program to dilate the shapes for the design of the rug. one of ashtis designs is shown below with labels for figures 1 and 2, and points x, y, and z. determine whether each statement about the design from the rug is true. select true or false for each statement. the center of dilation for figures 1 and 2 is point y. figure 2 can be dilated by a scale factor of 3/2 to form figure 1. figure 1 can be dilated by a scale factor of 1/2 to form figure 2. if figure 1 were dilated to form figure 2, figures 1 and 2 would have the same orientation.

Explanation:

Step1: Check center of dilation

The center of dilation is the fixed - point about which a figure is dilated. By observing the figures, the center of dilation is point \(Y\). So the first statement is True.

Step2: Analyze scale factor from Figure 2 to Figure 1

Figure 1 is larger than Figure 2. If we consider the distances of corresponding points from the center of dilation \(Y\), the scale factor from Figure 2 to Figure 1 is greater than 1. Let's assume a point \(P\) on Figure 2 and its corresponding point \(P'\) on Figure 1. The ratio of the distance from \(Y\) to \(P'\) and the distance from \(Y\) to \(P\) is \(\frac{3}{2}\). So the second statement is True.

Step3: Analyze scale factor from Figure 1 to Figure 2

The scale factor from Figure 1 to Figure 2 is the reciprocal of the scale factor from Figure 2 to Figure 1. Since the scale factor from Figure 2 to Figure 1 is \(\frac{3}{2}\), the scale factor from Figure 1 to Figure 2 is \(\frac{2}{3}
eq\frac{1}{2}\). So the third statement is False.

Step4: Consider orientation in dilation

Dilation is a transformation that changes the size of a figure but not its orientation. If Figure 1 is dilated to form Figure 2, the orientation of the two figures will be the same. So the fourth statement is True.

Answer:

The center of dilation for figures 1 and 2 is point \(Y\). - True
Figure 2 can be dilated by a scale factor of \(\frac{3}{2}\) to form figure 1. - True
Figure 1 can be dilated by a scale factor of \(\frac{1}{2}\) to form figure 2. - False
If figure 1 were dilated to form figure 2, figures 1 and 2 would have the same orientation. - True