In the figure below, Z is the center of the circle. Suppose that =QR20, =ST+3x2, =ZU12, and =ZV12. Find the following.

Break Down the Problem

1. Identify the given information:
   • $QR = 20$
   • $ST = 3x^2$
   • $ZU = 12$
   • $ZV = 12$
2. Understand the relationship between the segments involved in terms of their lengths based on the properties of a circle.

Relevant Concepts

1. In a circle, if two radii (like $ZU$ and $ZV$) are equal, they imply that $U$ and $V$ are points on the circle.
2. The segments $QR$ and $ST$ are likely chords or relevant segments in context with the center $Z$.

Analysis and Detail

1. Since $ZU$ and $ZV$ are both radii of the circle, we can establish that:

   • $ZU = ZV = 12$

2. From the context, it appears we might need to introduce a relationship involving $QR$ and $ST = 3x^2$.

3. We can assume that if $Z$ is the center and $QR$ spans a certain distance, there may be additional relationships involving angles or the triangle properties depending on the chords.

Verify and Summarize

We will set up an equation if needed:

• Since $QR$ doesn't specify a direct correlation in terms of angles or positions related to segments, we can conclude that the relationships are independent unless further information is given.

Final Answer

Currently, without additional context about relationships between the chords, the problem appears incomplete. The lengths provided are $QR = 20$, $ST = 3x^2$, and both $ZU$ and $ZV$ are equal to $12$. Further relationships or equations may be needed to solve for $x$ and verify the remaining values.