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In this problem, we have two intersecting chords, \( AB \) and \( CD \), in a circle with intersection point \( B \). To solve for \( x \), we use the **chord intersection theorem**, which states that if two chords intersect inside a circle, the products of the segments of each chord are equal. That is:

\[
AB \times BE = CB \times BD
\]

In the diagram:
- \( AB = 4x + 2 \)
- \( BE = 9 \)
- \( CB = 4x \)
- \( BD = 8 \)

According to the theorem:

\[
(4x + 2) \cdot 9 = (4x) \cdot 8
\]

Now, let's solve this equation:

1. Expand both sides:
   \[
   36x + 18 = 32x
   \]

2. Bring terms involving \( x \) to one side:
   \[
   36x - 32x = -18
   \]

3. Simplify:
   \[
   4x = -18
   \]

4. Divide by 4:
   \[
   x = 4.5
   \]

Since none of the given options (6, 4, 5, 3) match, it’s possible there’s an error in setup or an error in the answer choices given.

Would you like further clarification or have any specific questions?

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### Related Questions:
1. How is the chord intersection theorem derived?
2. What other properties of chords in circles are frequently used in geometry problems?
3. How can this theorem be applied to find unknown lengths in other geometric configurations?
4. What would happen if the chords intersect outside the circle rather than inside?
5. How would this equation change if additional terms were involved in the chord lengths?

**Tip:** For problems with intersecting chords, always remember to apply the intersection theorem if two chords cross each other inside the circle.

Solve for x in a circle where chords AB and CD intersect at point B. Given AB = 4x + 2, BE = 9, CB = 4x, and BD = 8.

Learn how to solve for x in a circle with intersecting chords using the Chord Intersection Theorem. This geometry problem involves the chords AB and CD intersecting at point B, with lengths expressed in terms of x. Suitable for students in Grades 8-10.

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