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Given that the pentagon \( RSTUV \) is circumscribed about a circle, we know that the sum of the lengths of the tangents drawn from an external point to a circle are equal. This property will help us solve for \( RA \).

From the problem:
- \( VR = 12 \)
- \( VE = 4 \)

We need to find \( RA \).

Since the pentagon is circumscribed about the circle, the lengths of the tangents drawn from the external points to the points of tangency are equal. Therefore, we have the following equal tangents:

- \( VR = VE + ER \)

Since \( VR = 12 \) and \( VE = 4 \):
\[ ER = VR - VE = 12 - 4 = 8 \]

Now, if we look at the tangents drawn from point \( E \), we get:
\[ ER = EA \]

Therefore:
\[ EA = 8 \]

To find \( RA \):
\[ RA = VR + EA \]
\[ RA = 12 + 8 = 20 \]

Thus, the length of \( RA \) is \( 20 \).

Do you have any questions or need more details on any step?

Here are some related questions for further exploration:

1. What is the significance of a circumscribed polygon about a circle?
2. How do the properties of tangents from an external point to a circle apply to other polygons?
3. Can you generalize the relationship between the sides of a circumscribed polygon and the circle?
4. How would you prove that the sum of the tangents from an external point to a circle is constant?
5. How can we apply this concept to find the lengths of other sides in polygon \( RSTUV \)?
6. What are some real-world applications of circumscribed polygons?
7. Can this method be used for polygons with more than five sides?
8. How do we handle circumscribed polygons that are not regular?

**Tip:** Always look for properties and symmetries in geometric figures that can simplify calculations, such as equal tangent lengths in this problem.

Explore how to solve the problem involving tangents in a circumscribed pentagon RSTUV. Understand the properties of tangents from an external point to a circle and their application in solving for side lengths. Suitable for advanced high school students and geometry enthusiasts.

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