We can use the Power of a Point theorem to solve for $x$. This theorem states that if two chords of a circle intersect at a point, the products of the lengths of the segments of each chord are equal.

In your case, the intersection point is $P$, and the lengths of the segments are:

• For chord $AB$, we have:

  • $AP = 8$
  • $PB = x + 2$

• For chord $DC$, we have:

  • $DP = x$
  • $PC = C$

By the Power of a Point theorem, the product of the segments of one chord is equal to the product of the segments of the other chord. Therefore:

$AP \cdot PB = DP \cdot PC$

Substituting the values:

$8 \cdot (x + 2) = x \cdot C$

Now, we can solve for $x$. First, expand both sides:

$8x + 16 = xC$

Next, collect terms involving $x$ on one side:

$8x - xC = -16$

Factor out $x$:

$x(8 - C) = -16$

Now, solve for $x$:

$x = \frac{-16}{8 - C}$

Thus, the value of $x$ is:

$x = \frac{-16}{8 - C}$

If you know the value of $C$, you can substitute it in to find the exact value of $x$. If you need further clarification or have additional details, let me know!

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Further Exploration:

1. How does the Power of a Point theorem work geometrically in the context of a circle?
2. What happens if the chords are tangents to the circle at the intersection point?
3. Can we apply the same theorem if the two chords are secants instead of just intersecting?
4. How can we use the Power of a Point theorem in problems involving tangents and secants to a circle?
5. What if one of the segments of the chord is unknown, but we have additional information about the circle?

Tip: When solving geometric problems involving intersecting chords, always check if the Power of a Point theorem is applicable, as it simplifies solving for unknown lengths.