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In this problem, you are given a circle \(V\), where \(W\) and \(X\) are points on the circle, and you're asked to find the length of the segment \(\overline{VW}\).

Here's what we know:
- The measure of \(\angle WVX = 20^\circ\).
- The length of arc \(WX = \frac{2}{3}\pi\).

The key point here is that the angle given, \(\angle WVX\), is the central angle subtended by the arc \(WX\). This means that the arc length \(WX\) can be related to the circle's radius \(r\), which is the length of \(VW\), since \(VW\) is a radius of the circle.

### Step 1: Use the formula for arc length
The formula for the length of an arc is:
\[
\text{Arc length} = r \times \theta
\]
where:
- \(r\) is the radius (which is \(VW\)),
- \(\theta\) is the central angle in radians.

We are given that the length of arc \(WX = \frac{2}{3}\pi\), and we know that the central angle \(\theta = 20^\circ\). To use this formula, we need to convert the angle into radians.

\[
\theta = 20^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{9} \, \text{radians}
\]

### Step 2: Set up the equation
Using the arc length formula:
\[
\frac{2}{3}\pi = r \times \frac{\pi}{9}
\]

### Step 3: Solve for \(r\) (which is \(VW\))
Now, solve for \(r\):
\[
r = \frac{\frac{2}{3}\pi}{\frac{\pi}{9}} = \frac{2}{3} \times \frac{9}{1} = 6
\]

Thus, the length of \(\overline{VW} = 6\).

Would you like any further details or clarification on the steps?

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Here are five additional questions related to this topic:
1. How would the solution change if the angle \(\angle WVX\) were 30 degrees instead of 20 degrees?
2. What would be the arc length if the radius \(VW\) were doubled?
3. How do you convert degrees to radians in general, and why is it necessary in circle problems?
4. What is the relationship between the radius of a circle and its circumference?
5. How can you calculate the area of the sector formed by angle \(\angle WVX\)?

**Tip:** Always make sure to convert angles into radians when working with arc lengths and sectors, as formulas typically use radian measure.

In circle V, m∠WVX = 20° and the length of WX = (2/3)π. Find the length of VW.

This problem explores how to find the radius of a circle when given the arc length and the central angle in degrees. Learn how to convert the angle into radians and apply the arc length formula step by step.

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