To solve this problem, we need to find the scale factor of the dilation and the center of dilation that transforms circle $T$ into circle $T'$.

Step 1: Determine the Scale Factor

The scale factor of dilation can be found by comparing the radii of the two circles:

• The radius $TU$ for circle $T$ (blue) extends from $T$ to $U$.
• The radius $T'U'$ for circle $T'$ (purple) extends from $T'$ to $U'$.

Measurement of Radii

• Radius $TU$ (for the larger circle $T$) appears to be 8 units (from the origin to the point $U$ at (8,0)).
• Radius $T'U'$ (for the smaller circle $T'$) appears to be 4 units (from the origin to the point $U'$ at (4,6)).

The scale factor is the ratio of the radius of $T'$ to the radius of $T$: $\text{Scale Factor} = \frac{T'U'}{TU} = \frac{4}{8} = \frac{1}{2}$

Step 2: Determine the Center of Dilation

The center of dilation is the point from which the dilation is performed. Since both circles are centered along the $y$-axis, and the dilation proportionally shrinks each radius by a factor of $\frac{1}{2}$, the center of dilation should be at the origin (0,0).

Final Answer

• Scale Factor: $\frac{1}{2}$
• Center of Dilation: $(0, 0)$

Would you like further details or have any questions?

Here are five related questions that could expand your understanding:

1. How do you determine the center of dilation if the circles are not centered at the origin?
2. How can we apply dilation transformations in different coordinate planes?
3. What are the effects of a scale factor greater than 1 in dilations?
4. How do dilations affect the equations of circles?
5. How do we determine the coordinates of points after a dilation?

Tip: Remember that a dilation transformation scales distances from a fixed center point, so the ratios between corresponding distances remain constant.