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Find the value of
x in the following diagram.

The diagram you've provided shows a circle with a point \( V \) at its center. Two line segments extend from the circle: one from point \( A \) on the circle to \( Z \), and another from point \( B_1 \) on the circle to \( Z \). The lengths of these segments are given as \( 6x + 20 \) and \( 10x + 4 \), respectively.

Since both \( A \) and \( B_1 \) are points on the circle and \( Z \) lies outside the circle, these two lines are tangent segments to the circle from the same external point \( Z \). According to the **tangent-segment theorem**, the two tangents from a common external point to a circle are equal in length. 

Thus, we can set the two expressions for the lengths equal to each other:

\[
6x + 20 = 10x + 4
\]

### Solving for \( x \):

1. Subtract \( 6x \) from both sides:
   \[
   20 = 4x + 4
   \]

2. Subtract 4 from both sides:
   \[
   16 = 4x
   \]

3. Divide both sides by 4:
   \[
   x = 4
   \]

So, the value of \( x \) is \( \boxed{4} \).

Would you like more details or have any questions? Here are some related questions:

1. What is the tangent-segment theorem, and why does it apply here?
2. How can you verify that \( x = 4 \) satisfies the given condition geometrically?
3. What is the general formula for finding the length of tangents from a point outside a circle?
4. How do the angles formed by tangents relate to the center of the circle?
5. How would the problem change if point \( Z \) were inside the circle?

**Tip:** Always check that your solution satisfies all geometric properties in a problem, especially when tangents are involved.

Find the value of x in the following diagram.

In this problem, you will solve for x using the tangent-segment theorem in a circle geometry problem. The lengths of two tangents drawn from an external point to a circle are equal. This example covers key concepts in algebra and geometry, suitable for students in Grades 8-10.

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