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?

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.