Line segment XY is dilated to create line segment X'Y' using point T as the center of dilation.

Question

Point T is the center of dilation. Line segment X Y is dilated to create shorter line segment X prime Y prime. The length of T X prime is 6 and the length of T Y prime is 9. The length of X X prime is 2.

What is YT?

3 units
6 units
12 units
15 units

Answer 1

To find the length \( YT \), we can use the properties of dilation. In a dilation, the lengths from the center of dilation to points on the pre-image and image relate through the scale factor \( k \).

1. We know that:

   • Length \( TX' = 6 \)
   • Length \( TY' = 9 \)
   • Length \( XX' = 2 \)

2. The scale factor \( k \) can be calculated using: \[ k = \frac{TX'}{TX} \] First, we need to find out the length \( TX \). Since \( XX' = 2 \), we can express \( TX \) as: \[ TX' = TX + XX' \implies TX = TX' - XX' = 6 - 2 = 4 \]

3. Now we can calculate the scale factor \( k \): \[ k = \frac{TX'}{TX} = \frac{6}{4} = \frac{3}{2} \]

4. Since \( TY' = 9 \), we can express \( TY \) in terms of the scale factor as follows: \[ TY' = k \cdot TY \implies TY = \frac{TY'}{k} \]

5. Plugging in the values: \[ TY = \frac{9}{\frac{3}{2}} = 9 \cdot \frac{2}{3} = 6 \]

Therefore, the length of \( YT \) is \( \boxed{6} \) units.