line segment xy is dilated to create line segment xy using point t as the center of dilation. what is yt? 3…

Explanation:

Step1: Find the scale - factor

The distance from (X) to (X') is (2) and the distance from (X') to (T) is (6). Let the scale - factor be (k). We know that if we consider the ratio of corresponding lengths in a dilation, and assume the distance from (X) to (T) is (d_{XT}=2 + 6=8) and the distance from (X') to (T) is (d_{X'T}=6). The scale - factor (k=\frac{d_{X'T}}{d_{XT}}=\frac{6}{8}=\frac{3}{4}) (but we can also use another approach). Since dilations preserve the ratio of distances from the center of dilation. Let's use the fact that (\frac{XX'}{X'T}=\frac{YY'}{Y'T}).

Step2: Set up a proportion

We know that (\frac{XX'}{X'T}=\frac{2}{6}=\frac{1}{3}). Let (YT = x) and (Y'T=9). Since the ratios of the distances from the center of dilation for corresponding points are equal, we have (\frac{XX'}{X'T}=\frac{YY'}{Y'T}). Also, we know that (\frac{XT}{X'T}=\frac{YT}{Y'T}). The distance (XT = 2+6 = 8) and (X'T = 6), (Y'T=9). Let (YT=x), then (\frac{8}{6}=\frac{x}{9}).

Step3: Solve the proportion for (x)

Cross - multiply the proportion (\frac{8}{6}=\frac{x}{9}), we get (6x=8\times9). Then (6x = 72), and (x = 12).

Answer:

12 units