The problem provided involves determining the scale factor used in a dilation transformation. Here's the breakdown of the solution:

Given Points:

1. Initial coordinates of $T$ and $U$ are:

   • $T(4, 2)$
   • $U(0, 5)$

2. After dilation, the new coordinates $T'$ and $U'$ are:

   • $T'(6, 3)$
   • $U'(12, 11)$

Solution:

The scale factor $k$ in a dilation is determined by comparing the coordinates of the original and dilated points. We calculate $k$ by finding the ratio of the change in each coordinate after the transformation.

1. For point $T$ to $T'$: $k = \frac{x_{T'}}{x_T} = \frac{6}{4} = 1.5$ or $k = \frac{y_{T'}}{y_T} = \frac{3}{2} = 1.5$

2. For point $U$ to $U'$: k = \frac{x_{U'}}{x_U} = \frac{12}{0} \quad \text{(This cannot be used, as the original x-coordinate is 0)} However, using the $y$-coordinates: $k = \frac{y_{U'}}{y_U} = \frac{11}{5} = 1.5$

Conclusion:

The scale factor $k$ used in this dilation transformation is 1.5.

Would you like more details, or do you have any questions?

Here are some related questions to explore further:

1. What is the general formula for dilation with respect to a center of dilation?
2. How can we determine if a transformation is a dilation just by examining coordinates?
3. How does a dilation transformation affect the distance between points?
4. What are some practical applications of dilation in real-world geometry problems?
5. How does a scale factor greater than 1 affect a shape compared to a scale factor between 0 and 1?

Tip: When working with dilations, ensure that all points scale consistently with the same factor, as this consistency confirms the transformation is a true dilation.